What alien bits should I sell?
Do not sell alloys, elerium, or intact components. Alloys go into almost everything, elerium goes into lots of high-end awesome things, and stuff like Flight Computers go into the cool vehicles later. Damaged stuff is ok to sell, as you can't do anything with it. You will need lots of alloy and elerium, and there is never an amount of this stuff that is "too much" unless you are near end-game and your stores are overflowing in all the neat shiny end-game toys. And even then you probably shouldn't sell this stuff. How many high-end craft you build is a matter of taste. But the high end craft will perform better than our "mundane" stuff, even without the awesome high-end munitions. Gun parts would also fit here. Though if you can capture a lot of aliens you'll capture a lot of alien tech intact. You'll likely have to build some high-tech weapons on your own at some point, though.
In addition to that, you pretty much need 1 of everything for research, but you knew that already. Some items might be marked as having no research value, you can sell those. I'm pretty sure (Damaged) items are thusly marked, in fact. It may be these are the only items like that. The items will be marked as having "no research value/safe to sell" in the grey market sell screen, when you select them (it will be so marked in the description). As Kotekzot notes, you can also safely sell alien entertainment, surgery, and stasis tanks. I think I've seen all there is to see and I can't recall anything else that's safe to sell off hand.
Corpses can be used in certain construction, e.g. Muton corpses for I think Titan Armor, and floaters are used for the dodge power up I think. Also, council requests will occasionally come in for corpses. I sell some but I like to keep a decent sized stack in reserve and I don't sell anything Muton or "worse". It's worth pointing out that council requests are uncommon, so saving all your corpses probably isn't going to pay off. Council requests will probably be of much greater value than selling the corpses, to be sure (I sold 6 light plasma rifles for $1500).
As of Enemy Within, you can get gene mods from autopsies, in addition to the existing research benefits.
By request, here are the things you can spend corpses on researching/building. The list is not yet complete as I can't check every price in my longest game as I've already done a bunch of it. I'll update as I track the rest of the items down.
Cost in corpses (yuck!) of various items
- Outsider Shard - (1) Skeleton Key (but I'm not sure if you can sell these anyway).
- Chrysalid - (4) chitin plating
- Ethereal - (1) mind shield
- Berserker - (1) combat stims
- Floater - (3) dodge power up
- Heavy Floater - (4) advanced repair, (2) Advanced Flight
- Cyberdisk - (2) tracking power up, (2) Advanced Flight
- Drone - (2) Drone Capture, (2) Advanced Flight
- Sectoid - (3) aim power up, (4) Xeno-Biology
- Sectopod - (2) advanced construction
- Thin Man - (4) Improved Med Kit
New from Enemy Within:
- Thin Man - (2) Gas Grenade
- Chrysalid - (2) Needle Grenade
- Seeker Wreck - (4) Ghost Grenade
Cost in components for various items
- UFO Power Source - (1) Firestorm, (2) Elerium Power Generator.
- UFO Flight Control - (2) Satellite nexus, (1) Firestorm, (2) Alien Nav Computer
- Fusion Core - (1) Blaster Launchers
Try to be cautious when selling certain things.
Try your hardest not to sell alloys, elerium, or weapon fragments, as you'll need those to research and engineer weapons and armor. I've also noticed that with the addition of engineering purposes, certain corpses and wreckages tend to be requested from the countries that you support. Try not to sell the following corpses, as countries tend to offer you stuff you'll need to fight the aliens:
- Thin man
- Floater (very few of these by late game)
- Heavy floater
- Cyberdisc
- Sectopod (you usually get rewarded handsomely for these)
- Chryssalid (hard to kill and dangerous early in the game)
These corpses are used for the powerups:
- Sectoid - aim
- Floater - dodge
- Cyberdisc - tracking
- Chryssalid - chitin plating
- Berserker - combat stims
- Ethereal - mind shield (very important by late game)
Other corpses don't get used as much so feel free to sell those when their bodies start to pile up...