The roulettes all unlock once you have at least two items from the list on there. So you'll have the daily lvlling roulette unlocked. To get the lvl70 you will need two 70 dungeons, Ala Mhigo in the MSQ is an easy one and then Drowned City is also in the MSQ. The lvl50-60 roulette should just require picking up one more lvl50 or 60 dungeon. Alliance you will need LotA and ST from the crystal tower series - the unlock quest starts in Mor Dhona. For the daily raid you'll need the first two Alex unlocked (Idyllshire), and you should have the daily trials unlocked already.
Beast tribes all unlock a bit differently. in HW the Vath are in Dravanian Forelands and the Vanu unlock at the Vanu settlement in Sea of Clouds. The Ananta in SB take a little more work, there are a couple of quest lines in the first map you need to do. For the turtles in the Ruby Sea the unlock quest is short and starts in Isari.
Squadrons start by simply ranking up in your GC, if you are at max rank you can get already then there will be an unlock quest at the GC HQ. You get members of the squad randomly by challenge log rewards. You will need to lvl the squad up to 20 before you can start doing dungeons (command) missions with them.
Hunts if you haven't unlocked those are in Ishgard and in the far east city I forget the name of right now

Look for the purple Q+ icon around the inn in both cities. The rank 1-3 bills all give exp. The elites are B ranks intended for lvl60/70, and just give gil.