It's not so much random groups, I'm very much a solo player of games so I've done most of the story alone and now am spending a lot of time doing missions and bounties alone to up my levels. I'm not so keen on pvp multiplayer, there is the crucible which is teams of people verses each other for various things like capture but I'm terrible. But I see other people with hours in there!
Easiest way to play with others is to join a clan or group on the destiny site or app. I think the app is great, there's a tower map, your own character profiles, easy access to the forums. I'm in a clan but I've never played with any of them as most seem to be ps4 and I'm ps3. But if I find a better one I'll join it. Each clan and group has a forum type thread thing, I don't know how to say it, but it's a list and it's basically things like level 22 warlock, Xbox one, looking to raid tonight 7 bst or whatever. And you add them, and join a fireteam of up to 3, and do story missions, raids, strikes, patrols.... Lots to do.
If you happen to be ps3 let me know if you get it and we can do some stuff, I've hit 25 and I'm struggling to get higher until the cap resets. You can earn up to 100 vanguard marks and 100 crucible marks a week, I've hit vanguard limit and I don't crucible! You use marks to buy better gear to level up. It's a convoluted system! So I'm mucking around until it resets and I've started a new character.
Oh and other way to find people is just to walk up to them in the tower, the social hub, and interact, or even on patrol or in mission. I was just patrolling earth tonight and got two friend requests, so I'll do some stuff with them tomorrow. You can even jump into other peoples strikes etc- if they allow it, you can set it so that can't happen.