I worked in Yellowstone this past summer, so I'll offer some advice.
Anywhere that you fly into that's moderately close to Yellowstone is going to be expensive. I'd just pick a city based on where in the park you're going to concentrate your time. Bozeman, MT is only 1.5 hours or so from Gardiner, the town at the north entrance of the park. West Yellowstone, MT is at the west entrance. Jackson Hole, WY would be the city for the Tetons and south entrance. Cody, WY is 1.5 hours from the east entrance. In any case, expect flights to be from $400 to $600 unless you find a killer deal.
As far as not renting a car... well, that's a challenge. The park is huge. If you're a bike rider, you could go that route, although it can be tough riding with a lot of distance between outposts. If you want to get involved in a huge hike in the west part of the park, I guess you could just hike in, do a big loop of some sort, then come back out the same way, but that would be logistically and physically difficult. Another option that always exists is hitch-hiking. I found people to be quite friendly in general, although there's always some risk with that.
In terms of time... if you're going to go before late-May or June, know what you're getting in to. Potentially a lot of snow, quite cold temps, big storms, tons of muck in the valleys, etc. Unless you have a lot of experience with that kind of thing or are planning to stick to the somewhat warmer and drier part of the park around Mammoth, I'd discourage anything earlier than mid-May to June. Do your research as to what routes should be ok by then, i.e. not covered in deep snow, having a foot of standing water everywhere, or intersecting with stream/river crossings that would be impossible at that time.
I hope you have a great time, but plan carefully and remember that it's a huge, wild park.