I spent a little time in and around the park. I would highly recommend the North entrance as far as family friendly, affordable, fun, exciting and numerous options go. There is lodging and camping in Gardiner, at Mammoth Hot Springs and on down the river valley towards Livingston.
That end of the park has all the stuff around Mammoth, including the Boiling River hotsprings (which will probably be closed in June due to runoff), and about 20 miles downvalley from the park are Chico Hotsprings, which also has lodging.
As far as WW rafting goes, the Yellowstone through Gardiner is class III, then there is a nice, scenic stretch for 10 or so miles, then Yankee Jim Canyon, which is +III & IV at high water. Certianly a few good drops. This is also big-water style boating, I think its pretty comprable flows to the Snake, but much closer to the park. Not to mention the affordablilty factor.
There's an outfit in Gardiner called Flying Pig
who's guides I kayaked with several times a few summers ago. They're a great bunch of folks, are right in Gardiner and have a sweet put-in setup for the Yellowstone right from their shop.
Cody is nice as well, but I gotta say the Shoshone is way smaller, and you are farther from the park. Once in the park, its a long way to the geyser/hotspring basins, going around the lake. This end (east and northeast) of the park is where I have seen the most wildlife. Lots of bison, and seemingly the best concentration of bears, wolves, etc. If you end up there, Core Mountain Sports
is the shop to check with.
Jackson is just out of my price range, even to visit. Its quite a ways from Jellystone, all the way north to/through Teton first. And the Snake is 1/2 hour south, but is really a beautiful stretch of river.
West Yellowstone is also pretty nice, and not far from the Gallatin, but still probably an hr. drive. The bonus with this location is the nearness of all the fun geologic features (Old Faithful, Paint Pots, etc.)
Wherever you go, you're gonna have a great time. There is just way too much cool stuff to do up there.