I did this when I was a student, and I recently had to move over the private repos to bitbucket when it expired. Just stick to bitbucket for private repos.
GitHub is a pretty great service (IMO). Totally worth paying for. If cost is your number one driving force, then sure -- go with Bitbucket -- but I think GitHub provides a way better experience (for users and developers).
Bitbucket is great for us as it's per user pricing instead of per repository pricing that Github has.
Github is fantastic for opensource projects or people with a low number of repositories - It's handy if you only plan on having a couple of repositories (if you are a SaaS etc).