This is a GREAT question. At 47, I think mine is the first generation that literally grew up with video games. I remember Pong when I was 8, and the Atari 2600 came out when I was 10, so I've seen every system and type of game as it was released.
Like someone above said, I love adventure games. Broken Age is awesome for anyone looking for that type of thing, and the full version was just released this month and it's from the makers of Monkey Island, Maniac Mansion, Psychonauts, etc.
I've been gravitating more towards Steam games recently. Getting 8-10 hours of gameplay for $10-$20 is great. Plus these independent game makers have all kinds of cool ideas that aren't found in mainstream games.
I also like the Nintendo series of Mario and Zelda games. I also really like the Batman Arkham series. South Park Stick of Truth is ridiculous and hilarious, though I think it really requires you to watch the Black Friday / Game of Thrones trilogy from the show, which was season 17, episodes 7, 8 and 9. Those episodes are pretty much a prequel to the game.
I like anything from Blizzard. I played World of Warcraft for YEARS, and only stopped when my bf broke up with me, and playing the game reminded me of him - because we used to play together. But Hearthstone is beyond awesome. Anything Blizzard touches turns to gold.
Back in the day I loved Banjo Kazooie and Donkey Kong Country. I love Guitar Hero and Rock Band, and am glad that they're both coming back.
I really respect games that are genuinely funny. Humor is hard to get right, especially in a game.
I would say, the type of game that I don't like, is the Call of Duty type of games. I just never got into them. I don't find them to be fun. But I like almost everything else.