Hmm... I think I have several suggestion for this:
1) Multiplayer Online Battle Arena: Dota 2/LoL - these two are so famous by now I bet we all have heard about it. These games are easy to access, but it will be difficult to master so I suggest you only choose to main one.
2) FPS: Warframe - Warframe isn't necessary a pure FPS but it can still give you quite a decent fun time with it. So far it's one of the best free online multiplayer games so I think it's worth a try.
3) Survival: Evolve - Evolve used to be a paid game, but after its poor players base the developers decided to make it a free-to-play game and I think it's a nice game for those who prefer a true beast hunting experience.
4) MMORPG: There are tons of free MMORPG out there and I can't really give any suggestion so I can only share my played games: Blade&Souls, Dragon Nest, Skyforge. There are probably more but these are all I can remember.