I like the Harry Potter books too. So there.
I think one of the greatest similarities is that both Harry and Frodo are effectively alone in the world. Both are orphaned and are able to act outside the constraints that parents and immediate family may place upon them. Of course both are taken in by relatives and brought up, but I would place Harry's equivalent to Bilbo as being Sirius. The Frodo/Bilbo and Harry/Sirius relationships are fatherly, but not the same as that a son might get from a father; in many respects they are like older brother/younger brother relationships.
Harry and Frodo both also have mentors - in the form of Dumbledore and Gandalf who they eventually have to learn to live without, which is important again, as it is another stage in their ultimate independence. I cannot wait to see how Harry responds to the challenges thrust on him in the final book, but I wouldn't be surprised to see him 'fall' much as Frodo does, and almost succumb, especially remembering him wondering in the very first book whether he would have made a better Slytherin than Gryffindor.