Does anybody know where I can get information on the Hungarian branch of my family to add to my tree? I have ancestors on both sides that I'd like to add.

Do you have more details on the persons you are researching?
The usual practice of starting out with genealogical research in Hungary is to nail down the name, date of birth, birthplace and religion of at least one person in the tree. Then you can start looking at birth/marriage/death records.

Unfortunately, we have far less information available online than you do in the US.