For a heal, you need to play either of the DoM to level 4. Either one will give you a heal. I suggest you keep that on your bars no matter what class you play.

Other classes that have self heals are the hand to hand combat one(pugilist,sp?) and the marauder(axe). Second Wind and Bloodbath respectively.

Also you dont just start getting health back even though you aren't in combat. You need to switch in and out of passive/active mode. Sitting seems to get health back quicker as well. These are all out of combat solutions.

As for where to go or what to do, My suggestion is look at some of the online community sites put together specifically for that game. FFXIV Origin, Zam there are others, just google it. I pefer FFXIV Origin myself, I find the guides easy to read and a lot of help getting started.