Should I go to a general doctor if I think I have lymphedema?

Yes. If you have swelling that is more than usual and is not going away, then a trip to the primary physician makes sense. There are many reasons people develop swelling....Kidney problems, heart problems, lymphedema, venous insufficiency etc. It is important to have it checked out and know the cause.
Start there. See your general doctor first. Lymphedema is probably something they feel comfortable in diagnosing and treating. If they don't they can refer you to a specialist. Board certified phlebologists (vein specialists) are trained in lymphedema.