I am going to buck the trend and go back to the home depot 2x4. Pine is hard to turn and get a clean surface, but it will give you a feel for needing sharp tools and a light touch. Try to get the best finish you can on pine. Put away the carbide tools for a little bit. Learn to use a gouge, skew and parting tool. If you can turn pine clean, you will be able to turn just about anything.
Downed tree branches are good, but if you turn them while green it will be a lot different than turning dry wood. I love green wood turning, but not for pens. Dry fire wood is a good choice if you can cut it down to a reasonable size.
Best of all, find another turner. Most of us have wood to spare and they can critique your technique.