So, you want to play? That's good!
I will quickly list some learning resources: I'm not going to bullshit you here, this game is tough to learn. There are a lot of rules and English-language material is scarce. This Flash game is a nice, light introduction to the basics in English with the absolute essentials listed on the page. Note that you should do yourself a favor and memorize the Japanese numbers on the character tiles while you play with this: whether you're playing a videogame or on a real set, you will usually not have the luxury of having Arabic numbers on them.
Play around with that for a minute, then read this friendly tutorial to get all the basics down. Then come back to the Flash game. There will be a world of difference between playing against these AIs and playing against real people, but just concentrate on learning how the game works. A truly exhaustive set of rules is available here: I keep a printout of the site around for reference.
Reachmahjong.com is the biggest English-language community: there's a lot of good material on the site but the design's kind of a mess so you've really got to dig for it. Also, their basic tutorials confused the hell out of me when I was first learning. Read their advanced stuff once you already know the game.
Oh right, they're talking about Tenhou. When you have a handle on how the game works, consider diving into Tenhou, the best, simplest online mahjong client. Yes, the client is in Japanese, but it is easily navigable with the help of this thorough English documentation. Read all of that. Tenhou is free to play for as long as you want, but at a certain rank you're limited to the "economy" version, which is charmingly and intentionally bare-bones, going so far as to replace voices with blips and bloops. However, there is a premium subscription ($5/month: not a recurring subscription, can be bought by the month when you want it) that gives you access to a pretty Windows client where every piece of the interface can be customized or changed outright. I have a Psyduck setup going.
One last thing: I'm always digging through mahjong videos on Nicovideo to watch pro matches. The language barrier means I can't get much out of the commentary, but it's still very valuable to look at how pros handle situations.