Death Note - if you haven't seen death note, you haven't seen a good dramatic psychologist at work.
Angel Beats - Cute girls, with weapons. What more could you ask?
Sgt. Frog - My personal favorite, it's a great comedy, non-realistic animation, and if anything made for a child to laugh.
Fairy Tail - Big series, good moral, lots of filler...
Sword Art Online - one of the best newer animes
Squid Girl - cute, childish, and funny.
Fruits Basket - good moral
Baka and Test - cute and competitive
Kenichi - martial arts and fighting
Kaze No Stigma - all time favorite, great story for a fantasy work
Eden of the East (all 3 movies) - Not sure exactly what order they go in :S
Ground Control to Psychoelectric Girl - 3 episodes of a simple plot.
Kill me baby - Cute and childish
Soul Eater - Great story
Attack on Titan - Bit gory, but a great anime
Full Metal Panic - Similar to Infinite Stratos
MM! - Harem... and kinda weird.
Acchi Kocchi - childish but cute
Of course you can just google anime lists, most of these were just taken from my personal list's first page :S