Alright, so you want to start off with something pretty easy and gateway-ish. You're probably too old for standard gateway shounen with like 600000 episodes though.
For ghost in the shell I'd recommend "stand alone complex", the 2002 TV series, though the original 1995 movie is also good.
For mecha shows I'd recommend something more recent for starters. Macross Zero(4 ep OVA), Code Geass, or Gundam 00 are good gateways. You can go back and watch the older stuff if you decide you like it. The continuity switches all the time anyway.
For space sci-fi try Cowboy Bebop, Outlaw Star, Space Battleship Yamato(the recent remakes are pretty good), or Gundam Wing. Legend of the Galactic heroes is really good but it's got like a bajillion episodes and a plot super far up its own ass so it might not be great for starters.
As for generally good starter shows try Steins; Gate, Baccano!, Full Metal Alchemist: Brotherhood(64 episodes might be a bit of a commitment), Psycho Pass, or Black Lagoon.
Pick some at random, google them(or search them on MAL or Crunchyroll) and if they seem interesting from the plot summary have fun watching them.