Create a game plan!
First, choose a weapon. In general archery tags are easier to draw, but success rates can be as low as 5 to 10% on killing an animal. I would suggest using a rifle for your first hunt.
Second, choose a species. Pick one species to target.
Third, apply for points. The good units in most states you need points to draw a tag.
Here is what I would do.
I would hunt with a rifle because it is your first trip out west. Decide if you want to hunt high or low. I would put in for points in Colorado for deer, and Wyoming for antelope.( Approximately $40 for each point) Again most good tags cannot be drawn your first year.In Colorado you will need anywhere from 3 to 9 points to draw a good deer tag. Wyoming you need 1 to 5 to draw a good antelope or deer tag.
You can apply for any region in Wyoming and have a good chance a drawing a GENERAL tag. The only exception is region G and K. You will not be able to draw these tags with 0 pts. Another option would be to apply for a deer point, and buy a leftover tag over the counter. By doing this you would have 1 point for next year.
Buy some BLM maps, and make sure there is plenty of public land. Make some call to the biologists and wardens. Get all your ducks in a row before you apply. If you don’t draw a tag, you can buy some over the counter.