We currently suggest Play65 for online backgammon. They offer play for fun as well as real money games and are one of the largest and busiest online backgammon rooms.
Once you download the software, play is pretty straight-forward. They offer tournaments, single games and series play, and thankfully players are ranked by their skill level (based on past play). Explore the site for a few days in fun play beofre moving on to real money games and you'll be all set.
Also look for features like the pip count and school that make online backgammon such an unique and interesting experience. Pay attention to the pip count and your play will likely improve. It's a big help in doubling and strategy. Also be sure to visit our backgammon rules page for explanation of all the options offered.