Utah has many opportunities for hunting forest grouse(ruffs, blues) and chukars on public land. Wild pheasant are a rare bird, though the state does release birds on state land during season. Quail are possible on public land down south. Some sharptail and sage grouse possibilities in the northern part of the state with a special permit. Better sharpie hunting in Idaho, better Sage Grouse in SW Wyoming, all within striking distance.
So there are many opportunities, just need to research and burn some boot leather.
I hunted dove, huns, pheasants, blue grouse, ruffed grouse, sage grouse (never bagged one but flushed them), waterfowl, quail, and chukar when I lived in SLC. There's a lot of public land in that state but if you move to the corridor between SCL and Provo where most of the jobs are, you're going to have to do some driving to get to where the birds are.
If I ever have vacation time again I'm going back to southern Utah for the quail.
If you're passionate about pheasants be prepared to drive out of state. 99.9% of the pheasant hunting in Utah is on private land and unless you're family you're not getting permission to get on.
If I had to move back there tomorrow I wouldn't be sad. If you end up moving there PM me, I can share some more info.