As many of the other posters said (and me before in many other Utah discussions), you do need to rent a car. A key virtue of Utah is that the driving distances are relatively short and you have the flexibility to decide where to go each day. With regard to the skiing Park City is very likely to be sloppy in early April, as are Snowbasin and Powder Mt. The expectation the vast majority of April weeks is that most of the better skiing will be in the Cottonwood areas.
So in terms of overall convenience SnowbirdDevotee's advice to stay in SLC is correct. You also can get discount lift tickets to all areas in SLC but not in Park City. You can still drive up to Park City (less than half an hour) for dinners, apres-ski a couple of nights. If being on-site in Park City for non-ski activities is more important, do that but be aware your daily commute to the better skiing in the Cottonwoods will be close to an hour vs. half an hour if you're in SLC.