Here are some sites that I use a lot and would strongly recommend. Maybe others can recommend their own as well, here are my top 5:
1. FanGraphs Baseball - This is by far the best baseball site out there IMO. They cover so much and are expanding. Advanced statistics, projection systems, batted ball data, minor league data, and the articles. The articles fantastic, and they branch out to other sites without hesitation. It's extremely user-friendly as you can post your own comments and ask questions. You should have this bookmarked, this is #1.
2. The Hardball Times - This site is very similar to FanGraphs in that there's a lot of useful content on it. I'm not a big fan of the layout so that sometimes bothers me. One of my favorite tools here is PrOPS. They also have one of the most complete glossaries.
3. StatCorner - Simple, and amazing. They're limited (good or bad) to the best stats for pitching and offense (tRA and wOBA), and I love their layout. You can look at total team production as well as a very efficient way of sorting the leaders in their stats. The way they break their few stats down is incredible.
4. First Inning - One of the first advanced sites I began to use, but since FanGraphs took off it really has little use. However, I do love their batted ball charts (here) and they have adequate prospect info. It may be on its last years though, as it needs some reconstruction.
5. Baseball-Reference - Not a bad beginner's site, as the player pages layouts are nice and keep the traditional statistics in plain view. I use this mostly for things like splits, team stats, league stats, etc.