It depends on what you want to do. If you enjoy mining, you'll make no profits but you can basically do it with no effort.
PvE can be fun. You'll make decent money and it's relatively safe, and you can put in as much or as little time as you want.
PvP is by far the most fun. You'll lose money until you're good, at which point you'll make lots of money. Maybe 3 hours a couple nights a week is plenty (though I play more).
Regardless, as a new character you'll have to log on at least once a day, just for a couple minutes, to update your skill queue. Later on you won't need to do that, as some of the higher end skills take over a month realtime to train, so you can basically just set them and go on vacation.