1. Shoshone
Absolutely love their UA. Extra land with every city you found! Yes please. And exploration is my favourite, the boosts I can get from early ruins (even on relaively high difficulty) are delicious. UI is very nice too! UU comes a bit late for my taste, but I still enjoy it when it comes around.
2. Iroquois
Bonuses for not chopping jungle/forest? Sign me up! UB is very nice, particularly early-game. UA makes wars in jungle/forest almost a bit too easy? I haven't actually got much use out of the natural-wonder related bonus, but I like the flavour. UU is fun, especially not needing any strategic to build them. Mohawks everywhere!
3. Byzantium
I often put a lot of effort into founding/enhancing early so I can get the stuff I want. Byantium takes a lot of the stress/pressure out of religion while also giving you a great source of faith! (Even if you have to wait a little while for it.) Cataphracts are also one of my favourite UUs.
Honourable mentions go to (4) Indonesia: all-time favourite UU (magic swords with mysterious enchantments!), unique luxuries and lots of gold (I love trade). (5) Polynesia: great for exploration, can steal all the ruins on islands before others can get there, fun UU and UI. (6) Celts for the absurd yields you can get from a jungle start with Cernunnos, the Horned Stag, and having zero care about religious pressure on your cities. Take that Gandhi!