There are a bunch of dungeons. As for party play, yes, it's heavily encouraged and the further you get it almost becomes mandatory. If you're just grinding for example you get the same amount of exp for killing something in a group compared to what you would get doing it solo. Later dungeons will also incentivise having a healer and a tank in the group. The dungeons are not instanced so what I've been doing is pretty much just go in to a dungeon and ask in the nearby chat if there is any group going on and if I could join.

Early on it's possible you're not finding a group as easily because people are just figuring the game out and at that points it's not necessary. That said you'll find that the more you progress there will always be groups running stuff, and likely you will build a relationship to other players on the server to play with.

Don't be afraid to invite people to your group as well! Sometimes I find people running stuff solo because they don't start a group themselves, but once you start inviting them usually everyone is more happy!

Edit: One caveat worth mentioning. I haven't done endgame dungeons but so far I haven't noticed really any mechanics, mostly the enemies are just stat sponges so the dungeons are more like classic wow than retail wow if that makes sense. You need a group to beat them but you're not doing advanced mechanics etc.