Oh wow I can't recommend playing the Metroid series enough, it's been one of my favorites since I first played Prime 1 when I was 7. If you have a Wii maybe you can hunt down a copy of Trilogy, which has all 3 Prime games just with Wii controls, though I personally prefer the Gamecube controller. Oh and if you have a GBA or care to emulate, Metroid Zero Mission's a great first sidescroller for you.
As for series I would like to try, I always wanted to play the Disgaea series, Persona 4, Nier Automata, and some strategy games like Total War and Nobunaga's Ambition.