for the most part, you get weapons from just doing stuff. strikes, quests, Crucible etc. I forgot when you'll have access to Banshee the Gunsmith at the Tower, but he would take Weapon Parts that you get from breaking down unwanted gears and give you random roll weapons. but I would suggest you to just grind strike/Crucible for a while once you've finished the main storyline and you'll get plenty of gear. and other vendors like Zavala and Lord Shaxx would also accept tokens you get from doing strike or Crucible in exchange for gears that include both armors and weapons. but again, I'm a little hazy on when you'll have access to them.
as for quests and stuff, a lot of them are just small story pieces but you'll still often get something out of them. if you have a yellow color quest, those would be for exotic weapons, so give them a go if you want some cool gear. thou I would suggest you to not actually finish those quest by picking up the weapon just yet and wait until you're 900+ light to save on the trouble of keep on upgrading them. I'm pretty sure you can just grind strikes and gear up to 1000 these days. you can look up some of the quests here:
Every Destiny 2 Exotic weapon and armour listed, including those from the latest Destiny season, Season of the Haunted.
www.eurogamer.net
I assume you just have the F2P base game right? if so, highly recommend you to work on getting Riskrunner the exotic SMG since it's super useful and still pretty decent in late games if you don't have anything better. Outbreak Perfected, Whisper Of The Worm are also really good weapons thou pretty sure you'll need some help for those 2. finishing the story will also give you some other exotic weapons quests. and of course Xur will appear every Friday and you can get whatever exotic weapon he's selling that week along with 1 piece of armor for each class. you'll need a lot of legendary shards thou if you want the extra exotic gear from the engram.
as for story progression, give this one a try and see if it helps.
Learn what order you should play all of Destiny 2's many campaign levels, from the Red War all the way to Shadowkeep and beyond.
www.shacknews.com