Some weapons are always useful, some weapons are always junk. Here is a list for a good weapon usage:
Shotgun: Works in (much to) close range only. Forget that thing.
Grenade Launcher: Can be used for destroying of bug-holes. The weapon has more ammunition than the rocket launcher, but it is weaker. This weapon fires in a ballistic arc, so make sure that your soldiers have a clear sky.
Rocket Launcher: Auto tracks a locked target. Stronger than the grenade launcher and your first choice for killing the big Tanker bugs or destroying bug-holes. You can also attack Plasma bugs, but then you need a lot of ammunition.
Nuke Launcher: As know as Hell-in-a-missile. A hit with that thing will kill everything in a wide area, so be careful. The weapon is good in killing large bugs like the Plasma bug or the Brain bug, but don't try to kill a Tanker bug with the Nuke Launcher. Sure, the Tanker will die on impact, but the explosion will make a barbecue of your soldiers, too. Another application area is the destruction of well guarded bug holes.
Flame-thrower: A defensive weapon. Sure, a single flame-thrower is not a thread, but a whole Squad with flame-throwers, formed up in a line will last quiet a long time. The flame-thrower is a range-effect weapon, that means that the liquid fire hits all the bugs in the area. Besides that, this weapon has a lot of ammunition. Unfortunately the weapon range is quiet small.
Mines: These little toys are placed by the engineer. Nice, but not necessary.
MIST-Trooper: A soldier with clocking ability and a sniper-rifle. The rifle is useless against big, armored bugs and killing normal bugs is a waste of ammunition. Primary usage of this soldier is to kill well guarded Brain bugs (when you have no more air strikes or the Brain bug has air cover) or to clean up the last Hopper bugs.
Smart Rifle / Chain Gun / Light Rifle / Light Gun / Plasma Rifle / Plasma Gun: The normal bug hunting tools. Each kind of bug is very vulnerable to one of these weapons (exception: Tanker). Equip your squads with a mix of these weapons and the missions would be easy.