Tourist/Visitor/multiple entry Visa?

I met a girl from Kunming, China when she was visiting San Francisco as a tourist. We kept in contact and I plan to visit her in February and possibly more often after that. I am Chinese born but a naturalized U.S. citizen. I've visited PRC twice as a tourist within the last ten years. When I visit her, I plan to stay with her at times at her home and other times we will be traveling around PRC. Do I apply for a tourist or visitor entry permit? Invitation letter needed? Can I or should I apply for a multiple entry visitor/tourist type entry permit because I will probably want to visit her numerous times within the next year? Thanks, Mike