This is a quick guide to what each weapon is good at. You can see the full weapon stats here:
https://docs.google.com/spreadsheets/d/1YN58lfYltnGPgYfW3b9QcEuaso_ZcsEzYkVTdz0nhbw/edit#gid=0

SMGs are the new, better Assault Rifles. They lose 1 damage and lose 20 aim beyond yellow-move distance, but in exchange they get +2 Mobility, Run and Gun (!), and +20 aim within blue-move distance.

Shotguns are ONLY good for killing low-armor enemies at close range, even if you give them Armor Penetrating bullets. However, they do have the highest damage in the game.

DMRs are the best 1-move weapons at ranges beyond blue-move distance, but as a long-range weapon very weak closer than that. Sniper Rifles are basically the same except without the ability to move and fire in the same turn and with 2 more damage. Which is a lot, but not never worth it if moving would let you flank an enemy.

SAWs have terrible stats, but the Bullet Storm ability lets them take two shots for two actions, giving them the highest damage in the game (assuming you don't need to move to get a clear shot). However, if you already have an ability that lets you shoot twice, the Shotgun will do the most damage, or against armored enemies, the Heavy Cannon will do the most damage. But keep in mind that you usually need a way to make up for the aim loss on cannon/saw.

As the game says, Laser weapons pierce armor making them good against medium-armored enemies, Gauss weapons don't making them good against low-armor enemies (or enemies with shredded armor), and Plasma weapons shred armor making them great at killing Sectopods, and decent against low-armor enemies. Some particularly good weapon+tech combos are Gauss Shotguns (high damage is multiplied on crit, and shotgunners stack crit), Laser Pistols (Guaranteed 3 dmg for free with the right skills), and Plasma SAWs/pistols (multiple shots = tons of shred).