Now, you might think the answer I'm going to give you is already obvious because I'm using GiHub right now, but it's not. Both GitHub and Bitbucket offer great Git services, but each has its own features and pricing plans. In the following... thing, I'm going to compare the two and then offer a final solution that should work for most people.
TL;DR: Both. Use GitHub for open source and public repos (you'll spend most of your time here) and Bitbucket for private repos. But, sign up for GitHub first, then import account into Bitbucket. Also, check comments for updates. P.S. I personally prefer GitHub.