You can get to Park City from the airport easily -- no car required. Uber/Lyft/several limo services provide lots of options.
Depending on where you are staying, Park City can be pretty easily accessed on foot for almost all your needs.
Deer Valley is the only Ikon resort in Park City; PCMR is Epic.
The other SLC resorts are Ikon: Solitude and Brighton are in Big Cottonwood Canyon, Snowbird and Alta are in Little Cottonwood Canyon. Solitude and Brighton are smaller, but (generally) less crowded. Snowbird is famous for harder skiing, and Alta is the old standard (and beautiful). Personally, I've grown to prefer Solitude/Brighton because they are more laidback, but if this is the first time, you might as well get in the more famous Alta and Snowbird. I bet most people on this board will point you there.
Getting up to Alta and Snowbird in LCC can be a pain; if you are willing to pay something, there is really nothing like staying at Alta or Snowbird and skipping the often painful commute up-canyon. If you decide to stay at the resorts, you can get a ride to them. If you decide to stay out of the canyon, you probably will need a car; public transport down in SLC is not really THAT easy or friendly.