Flight
You'll most likely want to fly to Bozeman, unless you have plans to extend your trip to further reaches of MT. Bozeman is closest to Livingston, Yellowstone, Big Sky and pretty much endless access to incredible public land. Bozeman is about 15 minutes east of the airport. Livingston is about 40 minutes east of the airport, and the venue is abut 15 minutes south of downtown Livingston. The only reason you might fly to somewhere other than Bozeman is if you want to check out the Missoula area (also a cool town, about a 3.5 hr drive from Livingston) or Glacier National Park. If you want to go to Glacier (which you totally should, it's epic), you should look into flying to Kalispell. Kalispell is about a 6 hr drive from Livingston. You could also fly to Billings, but it's further from Livingston than Bozeman and substantially less cool of an area...so probably don't.
Rental Car
I don't think there's any special direction on renting a car...except reserve one EARLY! Like 6 months in advance. The whole state ran out of rental cars last year and it was chaos. If, for your recreational time, you are planning on mostly hitting spots that are accessible by road--Yellowstone, the wedding itself, Bozeman & Livingston, any little car will be totally fine. The venue is up a dirt road, but is well maintained. If you want to do some intense hikes in the backcountry, you may want to consider renting a vehicle with higher clearance, as some of the roads to trailheads can be rougher.
Hotel
The Murray is definitely the coolest hotel in town. It's right downtown, and it has been famously occupied by cowboys, actors, writers and outlaws over the years. It's very nice, with the classic old school hotel vibes that I personally seek out everywhere I go. It's also right downtown (it pretty much is the epicenter of town), so if you want to check out the bar scene, it's an easy walk from everywhere.
Hotel
Expensive, brand new hotel...if that's your thing, or you have free points with Marriott, this is your place.
Hotel
We've never seen the rooms, but we recently walked into the lobby/bar/pool of this place and found it to be pretty nice, especially for the price. It's pretty close to the venue, and probably will cost hundreds of dollars less than other hotels in the area.
House Or Rental
You also know the drill...but again, book early!
House Or Rental
Pine Creek Lodge is the lodging option that is closest to the venue. You can rent cabins on-site (there is also camping). This is not a luxury spot by any stretch of the imagination. The cabins are located behind a popular bar/restaurant, which often puts on live shows in the summer. It would be a good spot for those who want to keep the party going! However, if you are looking for a quiet, quaint place...probably keep looking. Keep your eyes peeled close to the date for the live music schedule!
Hotel
Cool mountain setting, historic hotel and dining room plus cabins and (obviously) developed hot springs located in Paradise Valley.
Hotel
If you REALLLLLLY want to treat yourself, you would be hard pressed to find more lux than Sage.