There is so much to recommend in anime and i am mostly an old school anime watcher but i watch new stuff from time to time, these are some of my recommendations.
Lupin the 3rd, anything with lupin the 3rd really. If you're more into new stuff then watch the 4th season, jigens gravestone or the mine fujiko no ona series, the movie castle of cagliostro is a classic must watch.
Dagger of Kamui. Old ninja anime movie with awsome animation and a kick-ass soundtrack.
Re: Cutie Honey, a shorter take on the cutie honey series with awesome animation by gainax, if you like it then take a look on the 90 series new cutie honey (don't even get close to the cutie honey universe series).
Dirty Pair, watch the series, ovas or movies their all good sci-fi action comedy fun.
Bubble Gum Crisis, cyberpunk action with great animation.
Record of Lodoss war, for your fantasy fix, the series or the ova, both are good.
Kono Subarashii Sekai ni Shukufuku wo!, hilarious fantasy comedy about a boy that gets killed and end up in a fantasy rpg cliche world with the goddess that brought him there.
FLCL, comedy gold.
Gunbuster, gainax awesome animation in this mecha sci-fi ova.
Just a few of my recomendations