Renting a car from the airport from any of the major rental companies should be OK, go to a travel website (expedia, travelocity, orbitz) and do some price comparison, but book from the rental company website directly will save you money. No deposit is needed for dollar, national or thrifty which is really nice, and they are easy to cancel.
as for hotels, if you arrive late, spend a night at SLC, the hotels there are gennerally cheap. we stayed at a holiday inn last year for about $70. In the early morning, we drive north to the direction of Grand Teton NP, and spend some time at Jackson Holes for meals and some site seeing. Spend the second night inside Grand Teton, and rented one of the cabins, the cabins are really big (2-3 x of large hotel room) and facilities really good, the one we got had three beds which is great for a large family. We went rafting the following day and took a few trails and spend a second night there. The following early morning, drive to YS NP and stayed at four nights at different hotels within the Park. The pricest one is at Old Faithful Inn, almost $180 a night, but it is really worth it. Go to see the geysers in at dawn and dust is just unbelieveble. There are also the old rooms in the hotel which are much cheaper but they are the same as 70 years ago, try to see how people stayed in the past. Don't try to save money by staying outside the park, the time wasted is not worth it. all of the hotels in the park are officially sanctioned, so there is not much you can do to shave the cost. Fortunately they are very reasonably priced, you must reserve now, otherwise you may not find any since mid August is busy season. Spend all seven days at the parks, don't try to do too many things, YS needs more than that to see.
as for hotels, if you arrive late, spend a night at SLC, the hotels there are gennerally cheap. we stayed at a holiday inn last year for about $70. In the early morning, we drive north to the direction of Grand Teton NP, and spend some time at Jackson Holes for meals and some site seeing. Spend the second night inside Grand Teton, and rented one of the cabins, the cabins are really big (2-3 x of large hotel room) and facilities really good, the one we got had three beds which is great for a large family. We went rafting the following day and took a few trails and spend a second night there. The following early morning, drive to YS NP and stayed at four nights at different hotels within the Park. The pricest one is at Old Faithful Inn, almost $180 a night, but it is really worth it. Go to see the geysers in at dawn and dust is just unbelieveble. There are also the old rooms in the hotel which are much cheaper but they are the same as 70 years ago, try to see how people stayed in the past. Don't try to save money by staying outside the park, the time wasted is not worth it. all of the hotels in the park are officially sanctioned, so there is not much you can do to shave the cost. Fortunately they are very reasonably priced, you must reserve now, otherwise you may not find any since mid August is busy season. Spend all seven days at the parks, don't try to do too many things, YS needs more than that to see.