So, in terms of things you can directly go for an aren't just random drops, I'd say you need to get Whisper, Outbreak, the IKELOS Shotgun, a Blast Furnace, and a Hammerhead. The first two are missions you can go into pretty much whenever, although the initial Whisper portal is tied to public events on Io. IKELOS is tied to Escalation Protocol, and to get it you have to farm wave 7 on a week where the shotgun is available.
Other things to look out for are a cluster bomb rocket launcher, as that would be useful if you ever do Riven. Most groups will just want to do the cluster strat, though tbh any void rocket launcher would be fine. The reason void is preferred is because in that strat someone will use tractor cannon on Riven, which gives a 50% damage boost to void attacks. If you are lucky enough to get a group to show you Riven legit (best raid encounter in all of Destiny IMO) then you have a lot more flexibility. In that case you'd probably just use Hammerhead to have good add clear.
In terms of Blast Furnace rolls, you want a reload speed perk (outlaw / feeding frenzy) with either rampage or kill clip. It's the same for Hammerhead, but that doesn't have outlaw so anything with good reload speed and rampage is good. Both those guns are from Black Armoury, so you can farm those as much as you want.
A good shotgun to get until you get IKELOS is Hawthorne's. It's in the EDZ loot table, so hopefully it'll drop for you there or from Devrim. It's the same archetype as IKELOS but lacks the trench barrel perk, so it does less damage. It's not a huge difference since the shotgun rebalance though, and is actually fine for getting you through raids.
Sleeper is worth it too, so get that done. It's used less these days, but it's still viable in certain encounters. I know some people who still use it in Kali. I'd also recommend getting Loaded Question and 21% Delerium. Delerium is amazing, because you get a 33% damage boost for each kill stacking up to x3 and it lasts until you reload or switch weapons. It's very easy to get those stacks by tapping red bars, and when you have them it's a DPS monster.
So, that's some stuff to start on. Hopefully you'll get some good exotics while chasing that stuff. Basically though, with Whisper, Sleeper, Outbreak, and a good shotgun you can get through most of the raid content just fine.