I live in Jackson. As far as coming May 8th, It's going to be the start of summer season for a lot of people. Grand Teton Park opens to the public the 10th, and I'm not sure when camping starts, there's still a lot of snow out there right now.
Sick spots to camp? Most camping around is done in campground, granted most of them are quiet and spread out, for backcountry camping. a permit is required by the parks service if you're in GTNP, but if you're in the national forest, camp wherever the hell you want.
Start out with stuff near town;
-Curtis Canyon is sick. Right outside of Jackson, and you drive through the national elk refuge to get up there, $10 or so a night, but you can get away with a few days for free. amazing views of the Tetons and surrounding range. (PIcture stolen from my fb friend)
-I'd avoid Kelly campground, it's pretty much a big trailer park.. Avoid Kelly hot springs too. They're gross.
Moving south of town (towards Hoback)
-Granite Hot Springs/Campground sounds right up your alley. It's about an hour south of town, has a huge, spread out camping area, lots of great hiking nearby. The hot springs are weird though because it's a natural spring that settlers in the '30s built up into a pool. Cost like $5 to get in but so nice after a hike. Lots of backpacking around there too, but it's a common area for Mtn Lions, so be careful.
Heading north of town into the park. You'll pass a few lodges. (Dornans, Jenny Lake). You'll be directly under the grand at this point, definitely check out Jenny Lake, and I'd reccomend inspiration point. Few friends hiked it last year at sunrise (picture below). Backcountry camping is everywhere up here, and all the rangers have great spots set up, just make sure to get a permit. You can camp in any of the campgrounds for fairly cheap though.
Also check out Phelps lake, I'm not sure if the ice will be melted by then but later in the summer it's an awesome place to chill/camp/cliff jump.
Picture of the GF on the jumping rock from last summer.
I work at a resort a little further up the road. Right on Jackson Lake. But we don't open until the 10th.
As far as in town Jackson goes, the town square will have pretty much everything you'd want to check out on a quick trip.
Bars. (Best at the top to the worst)
-The Rose/PInky G's
-MIllion Dollar Cowboy (Tourist trap, but you have to at least get a beer and sit on the saddles)
All of the restaurants around here are pretty pricey, with the exception of Pinky G's (PIzza/Bar food) but the quality of food here is great. This will still be "off-season" technically which means 2 for 1 meals and cheap drinks. Check out Snake River Brewery, Local, D-O-G burritos, pearl st bagels..
Teton Village will just be opening for the summer around the time you're here (Don't quote me on that, it changes every year).
You can ride the tram to the top, or if it's clear, hike to the top and ride the tram down for free. Check out the Mangy Moose (Bar/Food) and Teton Thai opens up the 6th!, best Thai food in town for sure.
If you're willing to drive up into south Yellowstone, then I'd check out Flaig Ranch. There are some awesome, off the beaten path hot springs that will be great early May.
Let me know if you have anymore questions or want to meet up/get a beer. I'll buy the first round since I won't start work until the 13th.