The 5 basic weapon categories(Rifle, Sniper, Sub, Pistol, and Shotgun) are unchanged, but you get a lot more of each of them. You can now carry one of each of them provided you have enough carry capacity.
Heavy weapons are still around, but they are temporary pickups in certain missions.
They dropped the weapon research mechanism as it worked in ME2.
Now each weapon has two upgrade slots similar to ME1, but much more streamlined. You can also upgrade each weapon individually. I.E. Upgrade a Mantis I to a Mantis II to get slight stat bonuses. This can all be done from the Shuttle Bay on the Normandy.
Anything that burns targets works well; Incendiary Ammo and Incinerate weaken armor well and work on most targets. Cryo Ammo claims to weaken armor but that's only on chilled or frozen targets, and most targets that use armor instead of Health don't freeze, and they can be hard or impossible to even chill. Brutes/Banshees for example don't appear to ever be chilled.
Warp effects can help if you're a Biotic, as it also degrades armor. Most Biotic powers don't work very well otherwise, though Singularity does wonders if the target can actually be lifted; many armored targets can't.
For weapons, equipping the Piercing Mod and the Damage Boost mod both help; the piercing mod does more damage to armored targets.
Sniper Rifles work well, generally anything that does it's damage in one single shot works well, so Heavy Pistols can also do the trick. SMG/Assault Rifles don't work very well, and Shotguns don't seem that great.
Best Answer
As you can see by clicking on weapon links you can acquire them only in multiplayer.
From Mass effect official site