Yellowstone is a magnificent and unique spot and I do not want to discourage you, but it is very difficult to visit without a private vehicle not only because it is remote, but also because it is very large itself (almost 9000 km² or 3500 mi²— almost the size of Cyprus). Furthermore, unlike Yosemite NP, Grand Canyon NP, and some other parks, Yellowstone has no internal buses or other transportation system. This is all noted on the official website's Directions page. It is not hard to get to the park border, but to see anything more than the area around your lodge or campsite would require a lot of taxis or hitchhiking.
Ordinarily, I strongly prefer independent travel for a variety of reasons, but if you have your heart set on this trip, yours is a case where a tour with a guide or group might make sense, because of two points:
First, many Americans from the coasts, not to mention tourists from Europe or Asia, may not comprehend how empty the mountain West is, or how far points of interest are from one another. It is telling that the Park's official "Nearby Attractions" page has a category for cities within a 200-mile (322-km) radius. Even by this standard, SLC is not nearby; it is roughly 560 km or 350 miles, and just to get to the very edge of the park is a 5-hour drive in optimal conditions.
Second, it is already September, and given Yellowstone's remoteness and the extremes of its mountain climate, your options will be sharply reduced. This isn't so much because you might be snowed in or anything like that, but because the summer travel season in the US traditionally ends on the first Monday of September (Labor Day). The number of tourists drops very considerably and precipitously in the following weeks, and likewise establishments and services that cater to them will similarly be shutting down for the year.
There are a large number of concessionaires (i.e. people or companies officially licensed by the National Park Service to provide a service) which provide everything from guided tours to basic transportation in the park. Clicking through a few of the websites, I do see several that offer multi-day tours (with hotel or cabin accommodations), but it will be up to you to do some research as to who is still operating tours at this time of year and at what cost.
You may have more or better options if you can get closer to the park than Salt Lake City. Consider, for example, if you can fly into a closer airport:
- Jackson Hole Airport (JAC) in Jackson Hole, WY - American, Delta, Frontier, United
- Yellowstone Regional Airport (COD) in Cody, WY - Delta
- Yellowstone Airport (WYS) in West Yellowstone, MT - Delta
@MichaelSeifert points out that a little farther afield, but bigger of a town, is Bozeman, Montana (BZN), with scheduled air service from Alaska, American, Delta, Frontier, and United. Billings, Montana (BIL) and Idaho Falls, Idaho (IDA) are also listed on the NPS website's web page as serving Yellowstone, but these are much more distant and so the cost/benefit calculus will be more marginal.
Most service to these airports only runs in the summer and will have ended or will end very soon for the season. You can also look into bus transportation, though the only year-round bus connections to the West Yellowstone entrance appear to run from Bozeman, Montana; services from Cody and Jackson are seasonal.
