This is where I play around with the symbolism used in the show that           seems to have derived from 
historical           alchemy. Before diving into this,           it may be useful to browse through an alchemical text to demonstrate           just how fluid all the images and citations and discussions in here           are. I recommend           
Salmon's commentary on           the Emerald Tablet, which is one of the clearer texts I've found that           covers a lot of ground. Dense, but clear. Relatively speaking. It will           introduce a lot of the basic concepts.
Specific Symbols
Let's start with the particular symbols used most:           the crucified serpent and the ouroboros.
Crucified Serpent
Any fast google will tell you that the serpent cross means "fixing           the volatile", but that doesn't really tell you what it 
means so           let's unpack it a little. First, 
volatile in           general means "changeable". In specifically alchemical terms           it often refers to 
mercury,           which is certainly one of the most changeable elements alchemists worked           with. Thus, in laboratory terms, "fixing           the volatile" can mean rendering mercury by heat or other reaction           to separate it           into a stable form, or applying mercury as a solvent to a mineral.           In more broadly alchemical symbolic terms, it means           
a lot more.
Several of the substances produced by mercury reactions           are red. The color red usually stands for the active principle, in           alchemy. This may, from one direction or another, help explain why           Mercury, in code terms, often stands for Spirit. Spirit, active, animal,           mercury and red are all symbolically associated alchemical terms.
Some mercuric substances, such as calomel (mercury chloride), have also           been used medicinally. Notably,           mercury was also one of the treatments for syphilis, which, if it was           used in antiquity, may help explain its significance in alchemy as           a purifying agent. Efficacy against such a feared disease would certainly           make the reputation of any substance.
We find another connection to healing in the 
derivation           of the symbol itself, which appears to come from one of the biblical           tales of Moses, in which he nails a bronze serpent to a cross and displays           it to relieve sufferers of snakebite and/or plague depending on your           version.
So "fixing the volatile" generally means stabilizing the active principle,           something which can separate harmful and beneficial elements from each           other or even transform the harmful (pure active, too active) into           the beneficial (balanced active).
The symbol that Izumi, Ed and Al wear, however, is not the crucified           serpent alone. It is the crucified serpent winged and crowned. As we           can see in many 
manuscripts,           wings are used to mark progress or advancement of an alchemical solution           toward perfection. Crowns mark the final stage of a spirit or solution:           perfection, completion, ascension.
If we interpret snake = Mercury = spirit, which is a common symbol           chain, then the symbol can suggest that the final "rendering" of           the spirit, by death or enlightenment, will produce the pure, perfected,           incorruptible spirit that, in alchemical terms, tends to go along with           an incorruptible body. In this reading, the symbol indicates immortality,           the standard promise of the philosopher's stone.
One somewhat less classic way to read the whole symbol is as a           statement that when the volatile is fixed into a stable, presumably           healing, form, that 
is perfection. That to reach a stage that           is beneficial is either a) as close to perfection as we can come or           b) the only perfection we truly need to seek. My own feeling is that           this would be Izumi's interpretation. Certainly the fact that she wears           this symbol on her body, which we have no indication Dante does, suggests           that it means something significant to her. Another possibility, of           course, is that it is a reminder to herself of own hubris in attempting           human transmutation. It seems likely, after all, she knew that symbol           was associated with that transmutation in some way.
Ouroboros
The ouroboros is usually a snake or dragon biting its tail, though the           continual circle can be formed by other animals, such as 
two           birds. In any event, it is a consistent symbol of an endless cycle,           though the nature of that cycle is a subject of constant debate.
1 Once again, though, the version we see in FMA is altered. This time,           the dragon is winged and encloses the Seal of Solomon. This variation           is not unique, similar concepts  can also be found in the emblems           of           
Theosophy and           its offshoots such as Martinism.
2 It           does, however, indicate a different interpretation than the standard           ouroboros would.
Mclean mentions           that the dragon in general is a very double edged symbol, representing           the first alchemical stage (black, decomposed) when unwinged and the           last stage "the           spiritualising of the earthly substance" when winged.           Thus, there is a  double impact to the Homunculi's ouroboros: a winged           dragon biting it's tail could indicate a sustainable perfected reaction,           yet it invites contemplation of the unwinged dragon--and the Homunculi           are most certainly an expression of decomposition and of the cycle           of rebirth that an unwinged snake biting its tail would normally refer           to.
If you look through the 
image                archives, at The Alchemy Website, you will           find the Seal of Solomon pretty frequently. The two triangles that           make it up will be different colors, if the image is colored, often           blue and red or black and white. This is, as we will see in the Colors           section later, a common way of noting opposing forces such as water           and fire, or masculinity and femininity. Thus the Seal, which may show                the triangles interlaced, indicates the balance or combination                of opposing elements, up to and including the Above and Below.                In that last variation, the symbol necessarily indicates the balanced                unity of all that is, which can be taken as an aspect of perfection.
The Seal reinforces the notion of the winged ouroboros as an expression           of the final stage, of perfection. The symbol as a whole suggests eternal           perfection.
Thus, I would not say that the Sins' symbol and Izumi's symbol are opposites,           as I have seen suggested. They can be read as expressions of exactly           the same thing: perfection in balance, in unity, in healing. The Seal           and ouroboros, in particular, connect directly to Izumi's philosophy           of "All is One, One is All", and the incorruptible spirit           and body implied by the crucified serpent are certainly what Dante           is pursuing. They can also, of course, be read as different inflections           of the same concept, with the crucified serpent showing permanence           in stillness, at the apex if you will, while the ouroboros shows permanence           in motion, the cycle that is centered.
On yet another level, and taking into account the characters involved,           Izumi, Ed and Al can be seen as nailing down the runaway, destructive           cycle that the Sins embody and act out.           The appearance of the crucified serpent in the array Dante used to           contain and bind the Homunculi certainly suggest this.
On a somewhat disturbing level, the positioning of these symbols can           also be read as an indication that the Homunculi are a perfected state,           marked with the perfection-in-motion while they are active and bound           by the perfection-fixed. This reading suggests that the presence of           a soul creates imperfection, and that it is out of this imperfection           that alchemy is possible.
Colors
All symbols in alchemy are very fluid, and all of them have multiple           meanings, few moreso than colors. One thing we can say, however, based           on a study of 
colored           alchemical images, is that blue and red are often           used as opposites.
The most common blue, in the show, is military uniforms, and since           Ed's trademark is his red coat, and I can't help wondering if that           was entirely on purpose.
More interesting still, the illustrations seem to use blue and white           as interchangeable.  
Humberg mentions that white is associated with the feminine and red with the           masculine (or passive and active). On the other hand, "the crow's           beak blue as lead" rather indicates blue is a variety of black,           the crow being symbolic of Blackening or the first stage of transmutation.           We could, in good alchemical tradition, combine both interpretations           and thereby take the blue of the military uniforms as feminine/passive           in the sense of being bound to carry out another's will, and also black      in the sense of destroyed/destructive and taking the nature of death.
If the uniforms were a deliberate, symbolic choice, it adds a layer to           the fact that Mustang and Hawkeye are in civilian clothes when they           go to destroy Bradley. They have cast off passivity in both practical           and esoteric symbol sets.
The color combination of black-white-red is also employed in Ed's           clothes. Those are, in fact, the only colors he wears, and the balance           of them seems significant. Black, the color of destruction, and red,           the color of perfection, are both strongly represented. White, on the           other hand, the representative of recombination, of harmony, is only           barely present. The ensemble suits Ed's tendency to extremes, and also           suggests that he may be missing a step to actually get where he wants           to go.
Tidbits of Interest
Just some other things that caught my attention.
Stages
Humberg gives the three stages of alchemy as destroyed, recombined, perfected.           This seems to be fairly standard.
No one in FMA   uses those stages, though. The stages Armstrong           quotes to Scar are understand, destroy, recombine, which seems to leave           out the stage of perfection altogether. If perfection is synonymous           with the Stone, and the Stone, in the FMA-verse, is simply a shortcut,           a condensation of lives that the alchemist may draw on to circumvent           having to pay with his/her own, then this may be understandable.
Izumi, on the other hand, uses accept, understand, create (ep 27). Izumi's           version offers a hint that the ability to create is the distinguishing           factor of  human life/existence, particularly in light of how her           philosophy seems to derive from her experience of having and losing           a child. It is by her that we are reminded that the Elrics watched      Elysia being born, and she seems to feel this is significant to their development.
This also suggests why Izumi refers to the total perception of space/time/matter/energy           within the Gate as "a conjuror's trick": because to use that           perception to alter what the world is, alchemically, is to deny the           first step of her alchemy, acceptance. Ed it not wrong, really, when           he says it's the Truth; but it's an instrumental truth, a mechanical           truth. Truth, and total perception, are not wisdom. Izumi seeks wisdom:           action and instrumentality of will that are in accord with the shape           of what she has perceived, not seeking to alter that shape overall.
Texts
The text on the paper Roy hands Ed at the end of episode Eight, after           the whole "name           of Full Metal Alchemist" sentence, is taken from the end of Ripley's           
Recapitulation           of the Twelve Gates,      as egregiously allegorical an alchemical text as you're likely to find.            It's a fun read, really.
Little Observations
Both Izumi and Scar's older brother pay for their attempts at resurrection      with the body parts most proximal to the one being called back: Izumi's      uterus and nearby organs, Scar's brother's genitals.
The image on the Gate, an eye surrounded by a glory, is the Eye of Providence.           Often used to indicate the beneficial oversight of God, it is also           associated with the circled dot, which is sometimes a symbol of gold           and sometimes a symbol for the sephirot Keter. More on this in the           
Gate page.
The parts of humans identified by FMA, especially in the context of alchemy,           are 
inochi (life), 
tamashii (soul), 
seishin (will           or mind or spirit) and 
nikutai (body). The part that allows           alchemy to be done, the part the Sins are missing, according to Lust,           is 
tamashii.           The energy source of alchemy, as identified by Hohenheim, is 
inochi.
Footnotes
1. For an example of just how many ways the concept of ouroboros can           be taken and applied, see this 
conversation           thread on levity.com.
2. For the reference on Martinism, look a bit over halfway down 
this           thread. You may find it easiest to run a Find on "seal of solomon";           the second instance is the one you want.
Sources
My best source for things alchemical is 
The                Alchemy Website. This site           offers primary sources in abundance, both text and pictures, and their           archives contain a great deal of useful conversation between modern,           practicing alchemists. The site is not very systematically arranged,                quite in the alchemical tradition, but it is searchable.
If primary sources are too time-consuming for you, the 
Alchemy                Dictionary can usually give you a decent thumbnail                sketch. Keep in mind, however, that these entries are necessarily                abbreviated and, by that token, not entirely accurate.