I'll give you a few recommendations for a first-time player:
Microgaming casinos are probably the easiest software to use, and that's a great place to start.
Here are links to a couple of Microgaming casinos:
Mummy's Gold Casino
(I've been playing some good promotions here recently.)
Wild Jack Casino
However, Microgaming is mostly single player. (A few now offer a multi-player version, but I'm not impressed with it.)
If you want to play multi-player blackjack and interact with other players at the table, consider one of these Cryptologic-powered casinos:
(busiest multi-player tables)
Sands of the Caribbean
After trying out Crypto and Micro software, the next group I would recommend is Playtech software.
Here's a good one:
Before you do anything, be aware that your credit cards are probably not going to work for depositing. I recommend funding your accounts from your bank account via http://www.neteller.com
Don't be in a hurry to get going. Depositing into Neteller is free if you use the EFT method (not InstaCash). It'll take about a week for the funds to be available at Neteller. After that, you can easily deposit from Neteller into any casino.
I've named a few names here, but they are just a few among many reputable places you can gamble online. If you see a casino advertised on this site, it is because I deem them to be reputable, reliable, and fair. I also have a no-frills list of many good casinos separated by software, here:
For blackjack, Microgaming Vegas Strip Blackjack is the best choice among their many choices. Cryptologic rules are typical 8-deck, except they do offer a single-deck with a $25 minimum and OK rules (H17 though). Playtech's best game is Blackjack Switch, but that requires a complicated strategy. The best normal Playtech blackjack game is Blackjack Surrender.
I know this is a blackjack site, but I'll also talk a little about slots. I rarely play slots at a land-based casino, but it's not that unusual for me to play some online. There are three main reasons for that: First, I sometimes play bonuses that require slot play for the wagering requirements. Second, my experience has been that online casino payouts on slots are more generous than at land-based casinos. And third, I find the games online at Microgaming and Playtech casinos to be much more interesting than land-based slots.
Hope this helps, -Ken-