Naruto and Bleach were my gateways, but are really not that good and very genre specific.

Full Metal Alchemist: Brotherhood and Death Note were my "oh this is why people rave about anime" moments. FMA:B is a shouneny mystery thriller thing. Death Note is pure suspense all the way through, though a lot of people vastly prefer the first half.

Cowboy Bebop is a masterpiece, and is legitimately superior in the dub than the sub. Mary Elizabeth McGlynn should be paraded around the town for what an amazing job she did, in particular at a time when dub VAs weren't necessarily taken very seriously.

My all-time favourite is Baccano! It's an erratic suspense/thriller/comedy with the focus being around a train journey that ends in disaster.