Check out the Lodestone for FFXIV. The tabs on the left side give you the most comprehensive and up to date info on how to play the game.
http://lodestone.finalfantasyxiv.com/pl/index.html
Also do you have a USB gamepad? The game works ok with mouse and keyboard right now but unless you are a healer it tends to play more fluidly with a gamepad like a USB Xbox360 pad or a logitech pad (or if you can get the drivers running for it, PS3 controller can also work). One of the biggest changes coming in 2.0 is the overhauled UI for keyboard and mouse, the current one is fairly lackluster, it works but it takes some getting used to.
As for leveling, I'd recommend THM for leveling if you are a newer player. The first 10 levels aren't as fast as some of the other classes but once you get your first combo spell, you tear through mobs, especially in leves, like its no ones business. But try out a variety of classes and get them to lvl 10 or so and see what one feels like a good path. Make sure you get Pugilist (Hand to Hand) up high enough to get second wind, that ability will save your ass on just about any early level class. Its a low level Pugilist ability, like lvl 4 or 6. basically a free instant heal every 40 some seconds. Try to do the main missions when you can, they give decent gil at their completion plus they are the earliest parts you will see cutscene story's. The main missions are the stuff you initially start from the leader of the adventurer's guild at the city you start at (Last person you talk to after you sent out into the world, they will teach you about the leve system and lead you toward the main line story). The main story milestones trigger every 5-8 levels with about 4-5 of them before lvl 20 and then it starts to stagnate a bit. Once you get lvl 20 on a class you can do class quests for gil and EXP, at lvl 23 is the first Grand Company quest, and you can join a grand company officially at lvl 25, lvl 25 is also the earliest you can get a personal mount but it takes some time to get it. If you don't want to go through the long process to get a chocobo, you can get a goobbue mount at lvl 30 by just talking a single NPC (this is one of the "only in umbral era items which won't be available in 2.0").
You can also do crafting and gathering by equiping the weathered tools, gathering is probably the only class you can level and gain gil at the same time with little issue, not to mention you can often times get all the gear you will need for it from leves and sidequests if you are vigilant but most of the games current content comes from the combat classes so you should level up one of those initially, but the world is yours to do whatever you want or whatever you feel like leveling
.