Hello Bryoal and welcome to FFXIV!

The new player help forums might have some useful information for you, especially the leveling spots thread.

I recommend doing leves and grinding with a few other players or a full party for exp. Leveling a crafting class is always good, but you might want to level it with leves instead of buying the materials to save and make some extra gil. You can level different kind of jobs simultaneously, no need to focus on only one.

Unfortunately, you can't alt+tab with a fullscreen version of the game, only in the windowed mode. There's an unofficial fix for that though.

I wish you great adventures and enjoyable stay in Eorzea!