Now is the time to apply for antlered hunts this fall. I would suggest putting in for limited entry deer, Limited entry elk, and a general deer permit. The time to apply for antlerless hunts is in June generally. here is a link to where apply.
https://wildlife.utah.gov/2017-big-game.html
But you ask where does one put in for? That is only answered by yourself. Most likely with this being your first year you won't draw a Limited entry or general deer tag, however you will start building your points. Usually the more points you accrue the more likely you are to draw a tag. still there is a very small chance you could draw. It is going to take some time and effort to find where you would like to apply and many things can go into it. The following questions can narrow down choices.
How much time can I commit to the hunt?
How far am I willing to travel for hunt?
Do I like exploring new places?
Do I like staying in familiar places?
How much money can I invest in the hunt?
Do I enjoy hiking?
Do I enjoy driving dirt roads more than hiking?
How much effort will I put into the actual hunting while hunting?
Do I have a capable 4x4?
There are other questions but I think you get the idea.
Here are links to guidebooks for application and rules. They can tell you where the different hunt areas are and when the hunt date for certain species are in that area. I would narrow it down to 3-5 areas to further look into. From there you can use Google earth, hunting forums, and talking with other hunters to hopefully narrow it down to the area you want to apply. My favorite wildlife forum is
http://utahwildlife.net/forum/
Realize That asking about hunting areas is kind of a touchy subject to some. Take any advice with a grain of salt. generally I say you can believe about 75% of what other hunters will say. Many will not be truthful because they feel or fear that their "special" area on the mountain will now be overrun with hunters if they share it. Many will not share. Many will share where to camp and general area advice, but where to hunt in the area generally is not shared.
My advice is to start with a cow elk hunt. you can apply for those in june. They are generally a lot more laid back, have better success rates, and generally do not have the crowds that the general hunts have. You used to be able to draw them every year but lately it is a every other year affair.
Last but not least Youtube can be very helpful on not only how to hunt but more importantly what to do after you have made the harvest. From how to gut and transport to butchering if your so inclined.
hopefully that answered some questions.