Wow, I'd figured this was dead as dirt! 
Good campgrounds around here are Tom Miner and Pine Creek, both off US 89 down Paradise Valley, south of Livingston. 
Closer to Bozeman, Fairy Lake is nice. The road is awful.
No camping, but you should check out Madison Buffalo Jump.

Cool.  Thanks for the tips.  I have the DeLorme Atlas/topo map book.  Also a western US free campground book.  Both have good info.  I recently put AllStays app on my phone.  I'll tool around on that for a look also.  My stays in Bozeman have only been 4-5 days each time.  I take two days to get there and use three to get back.  I would like to explore the area, perhaps a fifty mile radius from Boze-angeles.  oops.  Did I say that ?  It'd be nice to camp a day or two at different locations.  Must be the gypsy in me .  Dunno.  The drive along the Gallatin river is nice.  I'm getting itchy to head back up to visit.  Maybe end of April but most likely in May.  I'm based east of the SF Bay Area.  It's 1067 miles from my door to hers.  My last trip there had me stay at a motel in West Yellowstone.  It was cold.  The town was dead , being off season, except for grocery stores and a coupla bars.  It was great, almost spooky.  My daughters BF and his dad are getting into jeeps.  I'll have to drag them out to the woods, set up camp, and conquer some mountains.  

edit,  When I go, it's three days and two nights there with four days and three nights back.  Just wanted to be clear on that.  It's basically a sixteen hr drive.