You can either enter the draw (too late this year) and earn a point if you don't draw or as gon4elk said just buy points for what ever species you want to hunt during the summer. You can put in for any species there is a hunt for in WY but you can only build points for elk, deer, antelope, moose, and sheep. You might want to look at some of the license fees for NR hunts though as some of them are really expensive, goat and sheep are over 2 grand, moose is about $1400. If you just want to hunt after college like you said then you're better off just buying points until then and you won't have to spend money on application fees.
Personally I don't see why someone in Utah would want to spend all the money to hunt WY (even an antelope tag is almost $300), but that's your choice. I'm not trying to get you to not hunt here, just seems like there is good hunting in Utah and it would be a lot cheaper. Are you just wanting to build points so you can hunt in really good areas?
Here is a link to the G&F site for license fees, look around on the site to learn more about when to buy points, apply for tags etc.
http://gf.state.wy.us/fiscal/license/licprices/default.aspx
One other thing to consider is nonresidents have to have a guide in WY to hunt in wilderness areas. You can get a resident to act as a guide but I'm not exactly sure how that works.
