I've just stared getting into anime the last couple years, and so far the shows I've enjoyed the most are:
Bleach: A highschooler accidentally gets the powers of a demon hunting Soul Reaper.
Naruto: This one is a little more juvenile than a lot of the stuff on the list, but I really enjoy it. Think Harry Potter with super powered Ninjas.
Full Metal Alchemist: Brotherhood: A pair of brothers who have been effected by a magical accident work for their government, while also trying to find a Philosopher's Stone to repair their damaged bodies.