The retro-arcade might not be the best place to learn these things (unless you have a knowledgeable guide with you). Not that you shouldn't go, but it's a hard place to learn anything.
It's easier to learn when you've got some more time and less pressure, so I really recommend buying a few, cheap, "gateway games" in various genres, that aren't too demanding on computer hardware. Lots of games are regularly under $5 at steampowered.com, humblebundle.com/store, or gog.com, and some have free demos. (But arcades, and "free to play" games, are mechanically set up to make you give them more money. It's easier to learn about the relationship between mechanics and genres without all that noise. That's why I'm recommending purchased games or games that are genuinely free.)
Do you have a PC or Mac? PC has a wider selection but lots of recent games are cross platform.
The following will give you a taste of a genre without overtaxing your computer, and these are frequently available very cheap. (Also, I tended towards lower-violence options when available, given your statement of preference earlier, and gave a bias towards "arty" games.) There are a lot of platformers here because it's a basic genre that you can find tutorials for; I also threw in a "walking simulator" because it's the easier kind of 3d game to make by far.
First-person puzzler: Portal
Puzzle: Sokobond
3d platformer: Psychonauts
Puzzle platformer: Braid or Fez
Precision platformer: Hmmm... VVVVVV? There's a free demo of that.
Metroidvania platformer: Aquaria
Roguelike platformer: Spelunky (still available in its original, free incarnation)
Roguelike: Crypt of the NecroDancer
RPG: Can't think of a good entry point here. Knights of the Old Republic, maybe?
Japanese-style RPG: Not a great selection on PC/Mac. Maybe track down Chrono Trigger by some means...
Action RPG: Torchlight
Japanese-style Action RPG: Recettear maybe?
Visual novel/Dating: Hatoful Boyfriend
Graphic adventure: Day of the Tentacle
Strategy: Hmm... Civizilation IV? The XCOM reboot? Any ideas?
Shoot-'em-up: Jamestown
Fighting game: Skullgirls (I thought Jamestown and Skullgirls did an admirable job of trying to train new players in genres that have become difficult to get into as a beginner)
Stealth: Hmm... Dishonored? Monaco?
Walking simulator: Proteus
Even cheaper option: Watch some "Let's Play"s or "Long Play"s on YouTube. Watching an insightful player play a game can be as illuminating, or more illuminating, than playing it.