Thane,
You can absolutely remove those branches. They may actually be suckers from a root stock but I can't tell from the picture. If they are then you will definitely want them gone as they are different from the tree you want to see growing. The small one you can just snip off close to the tree with some bypass cutters. the larger ones will require a saw. Make sure to use the three cut method and then after you remove the majority of the limbs then go back and cut up close to the trunk.
I wrote an article about doing this type of thing.
Pruning young fruit trees
It explains how to do these operations on fruit trees but it really does apply for most other trees as well. Take a look! Hope it helps.