I very much enjoyed both Arkham Asylum and Arkham City. For some reason, I lost my saved game in City, but I like it enough to re-play it anyhow. Looks great, story is solid, and the gameplay is a lot more diverse than it was in AA.
Great story, ambiance and feel too. They really give Gotham that gritty look and feel. Good colours and texture. No excessive light blooms or Lens Flares to try to over-compensate. It's a really well produced game. Story is solid too IMO. I highly recommend both Batman games to anyone who likes a solid action/adventure.
Though I admit AA is a bit more one-dimensional than AC, it's worth playing first because A) the story is righteous B) it's not super hard and C) the whole game should take you about 10 - 15 hours. You can finish the whole thing in a weekend if you don't go hardcore completionist and just focus on pwning the story through rather than doting over all the little challenges.