Hey popmuze - I'll keep it simple. Don't get stressed out trying to do Twitter/Facebook/G+/Goodreads and a million other things. Start by going where Beatles fans are, since Beatles fans are the target market for your book. Post and comment on their blogs, be helpful on their forums, make friends with them on Twitter, and so on. Even better, create something for free that Beatles fans will love (like a blog, website, video, tribute song, anything you can think of) and they will come to you.
But I guess you know Beatles fans better than I could - so you're the perfect person to work out where they hang out, and to reach them there.
Good luck!