Yellowstone: which part to go to, where to stay, what to do?

I’m going to be meeting my girlfriend in Yellowstone, me flying in, her biking in (part of a bigger trip). She’s going to be biking through Big Springs, but indicated that West Yellowstone and Flagg Ranch were all possible places to go.

So, I need advice on everything. Where should I fly in? What should we do there? Is it important to rent a car? Are there hostels or other less expensive places to stay? Any must-sees?

This will be around the second week of September, so hopefully not too crowded. We’ll see though.