When Heavensward launched I went straight to Ishgard and picked up AST - sounds like you had the same thing in mind.
For 30-50 I mostly ran dungeons, as FATEs are dull and healers get good queue times for most of them. You should also be running the Leveling and Trial roulettes each day for the bonus - if you don't have them unlocked, go and do that! (Unfortunately, as you noticed when you reached 50 on your first class, the Main Story roulette is dead so you're probably better off not sitting in queue for that. Far cry from the Mythology days, ha hah.)
There are some popular zones - on my overpopulated server at least - for grinding your face against FATEs, if that's more your style. Since you start at 30 you'd head straight for Costa Del Sol, move onto Coerthas Central after a few levels, and finally board the Northern Thanalan Bot Train for the last stretch.
