Well, I doubt that iTunes or app store will track your online activity. Though I might be wrong, I don't own any apple products to test it. However, if you just want basic anonymity for browsing and learning, then VPN/SSH should be enough. Just don't try to do anything stupid.
Remember this: once you will have a real need for advanced anonymity then you will know what you need to do. So for a start, I would recommend doing some research on virtual private networks (VPN) and SSH.