Arkham City has a demo you can try on Steam:
http://store.steampowered.com/app/200260/Batman_Arkham_City__Game_of_the_Year_Edition/

i think the games are mostly similar style so if you like it you'll probably like the others, too.

Yes, at least the Arkham series.
I've only played Asylum, City and Origins so far but I can tell you they're worth it. As a game (mechanics and level design) Asylum is superior, as a story Origins is the strongest but it's a bit of a glitchfest, and City is the best all-around product and experience but it lacks a personality when compared to the other 2.

In short, yes, they're really good and worth playing.

Basically a really nice mix of action and stealth. The combat is really friendly, and very cinematographic, something beautiful to watch, with just some button presses. Specially the counters! The occasional forced stealth I don't really like in any games, so I had to "tolerate" to keep playing the game.

Asylum has the best atmosphere/ambientation of them all;
City is a nice open-world experience, but feels blander compared to the previous.
Origins doesn't feel the same as the previous to some extent, but still plays nice sometimes.
Blackgate I don't have so I haven't played yet.
Knight (not included on this bundle) is awesome, but badly optimised. Still, loses the novelty of the combat mainly due to the excessive usage of the Batmobile (which is pretty fun). The investigative parts are really, really cool.

Dito. 😆😆

I think, that's just for the completionists. :)