Welcome,
It's beautiful here, but the hunting can be a bit competitive. The general season deer tags are on a draw, which has passed. There are typically leftover tags available for the northern region. Despite the amount of hunters afield, I have had good success in this area. Near the Jordanelle reservoir.
The archery deer tags are statewide. There are usually tags remaining after the draw. The archery tags also have extended areas which can be hunted 'til the end of the year.
The elk tags are purchased over the counter. I believe they go on sale june tenth. The elk units are broken into spike and any bull.
Run a search for utah dwr. You will find the Division of Wildlife Resources website. The site is easy to navigate. You can find a list of remaining tags, and where and when to buy. Big game proclamations are available at all the sporting good stores, department stores, etc. These contain maps and descriptions of all the hunting areas.
Hunting coyotes here is tough. They are so call shy they drop a rabbit and duck if it screams. I kill a few here every year. Mostly I just go for the sake of going. Nevada has a lot of dogs. There are a few red fox here, especially near water. They can be easier to hunt.
If you've never hunted elk and mulies before you're in for a treat. And the country is spectacular. I would be happy to show you some good places to hunt.