For playing: FIBS has been around forever, and if you download a decent client like 3dfibs or javafibs, it's not too bad. There are some annoying droppers who will quit to preserve their rating, but for the most part people are OK. You might want to ignore the chat.
Yahoo has tons of people playing for free.
I used to play a lot on TrueMoneyGames/XcitingGames, which has a great interface (and strippers!). Sadly most of the player base disappeared after a while. Not sure what it looks like these days.
I'd recommend against real money play at play65/gammonempire for the previous reasons, and some scandals involve well-known players being accused of cheating. Although there are tons of fish on there, the rating system and ridiculous rake means that the easiest way of winning is intentionally blowing your rating at small stakes games, which is against the rules, of course. I'm sure some people do make money there, but I hated the interface anyway. I made maybe $40 and left. Of course play money play is fine.
For learning: there are some excellent books on BG. Everyone points to Magriel's
Backgammon as the bible, though I've never read it, as I heard some things are wrong by today's standards...I understand it would still be a good starting point for anyone though. I started with Trice's
Backgammon Boot Camp, which I can't recommend highly enough--it's very well written, not dry like a textbook, with many important concepts explained through examples of various difficulty. You'll be surprised how much you improve just reading it once. Robertie's 501 problems is also good, though you'll want some experience first.
Download gnubg for sure, let it kick your ass a few times, find your biggest mistakes and try to figure out what you're doing wrong. No, it's not cheating, it's just that good.
Good BG websites:
www.bgonline.org has many opening move rollouts as well as a truly fantastic forum, with tons of discussions by some of the best players in the world. Stick (who runs the site) usually has an online match versus the readers going, though he tends to skip what he deems are "easy moves," which may not be easy to the rest of us.
www.gammonu.com is run by Kit Woolsey and has a lot of excellent content. Many of the archives at
www.gammonline.com are good too (see the annotated matches).
www.bkgm.com is a good general content site, with many articles and links, as well as archives of rec.games.backgammon. Do not, under any circumstances, attempt to read rec.games.backgammon today. It will make you sad.
Hope this helps! It's a great game, I like it (as you might have guessed). Sadly I am way out of practice since I've been playing much more poker and scrabble (heh). Honestly I was never an expert anyway