if you are new to them both then it depends on what you need. a free github account doesn't include any private repositories whereas bitbucket free account gives you unlimited private repositories.
this was the selling thing for me - I didnt want to initially pay for github as i was trialing it but i didnt want me code to be free for all as it was commercial. This lead me to choose bitbucket
I now use both (github for my OSS products and bitbucket for more locked down code) but I am moving my entire team away from SVN to mercurial as it was a simplier step to go to that git. From personal experience, it was easier to get to grips with Hg initially before git when I was using a DVCS for the first time. It didnt require the complex git install and could be used from within the windows cmd or powershell windows
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