I worked in the park summer of 07 as a wrangler at Roosevelt Lodge. I highly recommend you look into taking part in the cowboy cookout dinner that Roosevelt puts on each night through the summer season. You can either ride horses (one and two hour rides available, total time in the saddle prior to dinner will be more along the lines of 1:40 and :40 with a short ride back to the corrals after the meal), or ride in one of the chuckwagons that goes out as well. All total, we would feed between 180-210 people each night an all you can eat steak dinner.
Aside from that, within the park itself, check out old faithful, norris geyser basin, lake yellowstone, fishing bridge, mammoth hot springs, the upper and lower falls at Canyon, and the lamar river valley (nicknamed America's Serengeti). With the crazy snow year they are having, it's unlikely you'll be able to do much hiking, but the lamar and slough creek areas should be snow free by late June and can provide a lot of wildlife sightings. Spring runoff may have all the streams too muddy for fishing at that time sadly.
There's a chance dunraven pass (Mt. Washburn) could still be closed (it's looking to be a record snow pack year at the moment) and I'd guess the beartooth highway may not be passable either. However, if it is, I highly suggest you also drive the Beartooth Highway (HWY 212) from Cooke City to Red Lodge and eat at Foster and Logans in Red Lodge. It's an amazing drive with a chance to see some mountain goats. If it's closed, you could take the very scenic Chief Joseph HWY through Sunlight Basin from Cooke City to Cody.
For shear alpine and mountain beauty, the Tetons and Beartooths are superior to YNP in my opinion, so I suggest spending a good bit of time in those areas. In the tetons, if the tram is running, you can ride it all the way to the top outside of Jackson and get some spectacular views. Jackson is also home to some incredible art galleries and the National Elk Refuge. West Yellowstone is also a cool little town and has the Grizzly and Wolf Discovery Center which is great if you just want to lay eyes on a few, kids love this place as well.
If you want some other ideas, PM me and I'll see what I can come up with. One thing to note is that lodging, if you haven't already booked it within the park, is likely not available, as you'll be there during the peak of the tourist season. Your best bets will be Cooke City, Cody, Gardiner, West Yellowstone and Jackson for lodging if the park is full. You can also camp at the various campgrounds inside and around the park, but they could also fill up quickly that time of year.