Most commonly used primary weapons in crucible are exotic handcannons, red death, the messenger (osiris pulse rifle) and the mythoclast. Jewel of Osiris is also becoming pretty popular (osiris hand cannon). If you're the type that likes to rush and be offensive, use The Last Word or Mythoclast (Thorn isn't too bad either). If you like to sit back more, then Thorn, Hawkmoon (for PS players), Red Death or The Messenger are better choices.
Shotguns: felwinter, matador, party crasher, judgement vi. Try to roll for max range and either shot package or range finder. Don't go running after people with a shotgun as they still have limited range. Only use them if you can sneak up close to someone, or if you see people rushing you.
Rocket launchers are generally preferred because it's a quick kill. If you want to use an MG, you can possibly get more kills, but you have to be smart and play more safe. Basically you want to wait until everyone fires their rockets, then you should be able to string together some kills.
For rocket launchers, you want high blast radius and good velocity. Horseshoes with tripod is the ideal roll (IMO). Tracking is less effective in crucible because players move more unpredictably than AI enemies and can therefore easily dodge a tracking rocket (unless it's truth or gjallarhorn).
General tips:
-Try to learn the maps and where the ammo crates spawn. And pay attention to the heavy ammo announcement.
-When heavy ammo arrives, watch for enemies baiting it with their super. Stay close, but not too close to the ammo crate. Also try to wait for nearby teammates before you grab it.
-When shooting at people, try not to stand still... strafe back and forth.
-Use your radar often.
For Control, every map has a preferable 'side' in which you want to control. Once you capture that side, you want to hold that zone, as well as the next closest zone. Don't even bother controlling all zones unless the other team sucks. In most cases, just get two zones and defend them.
For any other team-based match types, you generally want to stay together.
In skirmish, don't respawn unless you don't think your teammates can get you, because revives give you points.
In rumble, find a nice little spot that enemies kind of frequently visit and try to hold it (close to ammo crates would be a good spot). Don't go running around the map. I play clash matches very similarly actually.
Clash, rumble or skirmish are probably the quickest match types and are therefore the quickest way to get marks. I personally recommend clash because there's a higher chance that you'll get wins, unless you're good at rumble or you're in a good fireteam of 3 for skirmish. A lot of the times I've played skirmish, there's a fireteam of at least 2 on the other team, making it harder if you're by yourself, because often times I get paired with noobs (which is why I usually only play skirmish with friends).
I'm no crucible expert, but that's how I generally play. When I first started playing crucible, my K/D was ~0.9-1, but once I learned the maps and got more experience, I raised that up to 1.4 averaged across my 3 characters. Which isn't great by any means, but enough to get flawless every week.
Plus my K/D would be considerably higher if I didn't play so much trials. In regular crucible, my K/D is usually around 1.6-1.7 or higher. In trials, it's 1.25-1.3.