The ammos work the same for the squad as they do for Shepard. What to use is up to your preference. Some of it will come down to what is available, and that partly depends on how much you try to max XP (since drops are dependent on level).
All things being equal, Tungsten VII does the best damage to synthetics, and Shredder VII does the best damage to organics. From a pure damage perspective it makes some sense to stockpile these when they start dropping (possibly also V and VI versions as well) and switch them back and forth for Shepard and the party dependent on the enemy you face on each mission. Armor Piercing and Anti-Personnel Rounds do the same thing, but have a lower bonus.
Of course, all things aren't really equal, and some of the niche ammos do make sense in some situations.
As Red said, some of the ammos prevent Krogan "second wind," and in theory should be fire (incendiary / inferno), cryo, or toxic (chemical / polonium / sledgehammer), and apparently proton / phasic (dunno why). However, if the Krogan is ragdolled they aren't supposed to be able to go into second wind and should just die no matter the ammo.
Going through the specialty ammo by class:
Incendiary / Inferno - not too bad, if you notice Incendiary VII does more damage than Inferno VIII, but IIRC Inferno VIII has a better accuracy penalty. Honestly if you want to be lazy equipping everyone with Inferno makes sense to me and I will do that sometimes, especially by late game.
Shield Bypass - This is where distinctions start to get muddy. The standard shield bypass ammos are Phasic and Proton. These deal reduced damage, but have relatively good shield bypass stats. Sometimes I might use them as a change of pace, but I personally think they don't make sense except possibly against a couple Geth enemies, and even then many times it makes more sense to use Tungsten. I am not sure if there is some break on some SR w/ Assassination that favors shield bypass. Supposedly prevents Krogan second wind (don't think I tested / noticed that but I don't run this often). Cryo Ammo is a special case in that it has more modest shield bypass, no damage penalty, but also has an accuracy penalty. I think it is a decent choice if you don't have a whole lot of other options, plus it is neat seeing stuff freeze and explode into snowflakes.
Cold Ammo - Touched on briefly above with Cryo. The "higher" level is Snowblind, which I never use anymore. The rate of fire penalty is a killer on ARs and Pistols, which are the weapons I favor. It might be ok on a dual Rail VII SR or SG. Just don't think about putting them on ARs or Pistols because there is no way to make up the DPS loss from the ROF penalty.
Toxic Ammo - Chemical and Polonium do toxic damage, which just means it is a barrier to health regen. The enemy basically has to waste regen ticks going through the toxin before they can regen HP. Your party can be affected by enemy toxic attacks as well (Rachnic, Thresher Maws, Creepers). Decent for preventing second wind on Krogan, ok against organics on side missions. Useless vs Geth. Sledgehammer is a special case which is supposed to deal a little toxic damage. More in a second.
High Impact - Hammerhead and Sledgehammer rounds. I actually find these slightly more useful on squadmate AR's for a little CC. Most of the decent SGs and SRs already have enough weapon force for ragdolling the annoying enemies anyway, but if you slap Sledgehammer on squaddie ARs they can do respectable CC against Creepers.
High Explosive - I think this is mainly a gimmick, but if you want a squaddie with no SR training to have a grenade launcher, you can consider it. I have never had a positive experience with Master Marksman pistol or Master Overkill AR's running HE compared to just running a more normal ammo. Can use on untrained SG if you want. I don't bother with it most of the time.
Radioactive Rounds - Some other rounds talk about radiation sickness, but they are really Toxic. The only one in this class is really Radioactive Rounds, which are an early game ammo that supposedly increases enemy cooldowns. The issue is that it only works after they have already cast a power, and I don't find it all that noticeable. But if you are doing side missions early game and don't have anything else, it is better than running Armor Piercing on a squad gun or no ammo at all.