Where you go kinda depends on when you go. The Park opens for fishing the last weekend in May; I think the same for the Madison below Quake to Lyon's Bridge. If your there in May/June you'll probably be hitting runoff on a lot of the rivers, but Madison (in the Park and above Beaver Creek)/Firehole/Gibbon should be fishable with good hatches. If you base in West Yellowstone you'll have easy access to those waters; not sure what kind of lodging you can get for $50-75 (Campfire Lodge, as noted, may work for you). My favorite shop is Blue Ribbon http://www.blue-ribbon-flies.com/ ; great shop, great people. Used a guide from them once and was pleased
Rivers in the NE end of the park will most likely be either too cold or in puking mud; Yellowstone outside the Park may be fishable.
Congrats to your son, and you.
Regards,
Scott
ps- looks like Wally put in his 2 cents while I was typing