Depends what you want to do and who is with you.
Lots of nice places to stay downtown, there is a brand new spring hill suites, the hotel jackson, white buffalo club...all of these are probably $350 plus a night, but within steps of the square. For a bit less but still down near square, Lexington hotel is clean a nice but not a luxury property, and the quality inn 49er is the same, but I would guess even they are in $250/night range. A bit further off the square the Hampton inn is clean and fine if you just need a home base and if the rate is significantly less than others can be a good move. Highly unlikely to find much for under $200 night in the area.
If you wanted to stay in Teton village the rates are pretty similar but it's a different feel, it's about 15 min from the square as a drive. Hotel Terra, Snake River Lodge, Teton Mountain Lodge are all nice and in the $300 night range.
For a bit different view and place to stay, Spring Creek Ranch is on a butte overlooking the tetons and the valley and the views are great and the facilities and service up there are usually excellent. Rooms aren't fancy but they are spacious, and I would guess $250-300 night. If you are a real high roller then stay at the adjacent Amangani hotel on the same butte for probably $900 plus a night.
Plenty of nice vrbo type options in the area as well. If you are in a tighter budget then maybe consider staying across the pass in Victor, ID or Driggs, ID at vrbo type places as the rates over there are much more likely to be under $200 night and the drive is about 30-45 minutes into jackson, barring traffic.
Peak season can certainly jam the area highways.
For things to do, it's your pick: hiking, Grand Teton, Yellowstone, mountain biking, horseback riding, shopping, wildlife and park tours (I have used brushbuck tours several times and all have been great) as well as float trips and whitewater rafting. For an easy hike I recommend the phelps lake loop or the Jenny lake.
If you or anyone going enjoys beer then there are a number of good breweries. In jackson, Snake River Brewery is the most popular and has some great beers. If you do get across the pass, stop in at Grand Teton brewery and their tasting room is small with little fanfare but it's very cozy.
If you have any interest in liquor, stop in and do a short tour as well as taste a few of the offerings that Jackson Hole Stillworks is doing, just south of town in Jackson. Currently they have a vodka and gin, whiskey is coming sooner than later, but tell the owners Travis or Chas that Clay said hello.
Food options could go anywhere for any budget and taste. Higher end, I like Trio (which is just off the square), Local (which is on the square), Bin 22 (for tapas and its just off square), and Cafe Genevieve.
If you must have a Mexican food fix then Merry Piglets is better than average especially for out of the southwest, most of the BBQ places around are serviceable, just know it's not much brisket on menus. For a good burger there is a place off the main drag, not on square, called Macphails, and while not cheap, the burger and fries are excellent. Snake River brewery serves up some really good mid range food too. One of my favorites is to stop by Local Butcher (just off square) and get a sandwich there (they have daily specials but I'm partial to their house smoked corned beef Reuben). If you wanted a bit nicer vibe and some seafood then Rendezvous Bistro by the Hampton Inn has some great dishes and almost always oysters. For a good upscale taqueria type place, hatch is great, and late night, pinky g's pizza is open.
For breakfast Bubbas BBQ, although I can't endorse them for BBQ, they have a large and well appointed breakfast buffet. Pearl Street bagels and The Bunnery are two other good options to try for anything from just a lighter muffin to a full on heavy greasy breakfast. If you must get donuts then try out Nom Nom donuts as it's a local favorite.
Feel free to ask, I know more than I should for not being a local, but while it's not cheap it's a great area to visit and see.