It depends on what Elk tags you are talking about. If its a general Bull Elk tag then they are over the counter, depending on the unit you choose you can hunt either spike only units or any bull units. There are only a handfull of any bull units and they tend to have low numbers and tough hunting to say the least, but with enough scouting and time spent learning the areas it is very possible to kill a bull. The spike units are basically all the limited entry units but all you can shoot are spikes unless you archery hunt then cows are legal as well.
Then you have your limited entry units. These units are the ones Utah is famous for. As far as time to draw they run anywhere from 2-4 years for the lower end units to 20 plus years for the upper end units. Your choice of weapons also can effect the amount of points it takes to draw.
The deer tags are run much the same way, general tags are draw only but you can draw these almost every year depending on the region you chose and weapon, and then the limited entry units take about the same point wise as the elk units.
Hope this helps.