Every year I meet folks that come out here to shoot a goat, for the most part it usually works out pretty well if they do there home work and and have an idea what to expect once they get out here. But in lot of those cases hunters that have never been out here don't have a clue how to hunt these critters in different areas of the state. There is more to hunting Antelope than just drawing and finding land to hunt. You had mentioned the areas around Laramie, some of it is rolling hills with Sage Brush not to bad for "putting the sneak on a goat" but allot of it has no brush or trees or even hills for that fact. The same goes for the areas around Cheyenne and a bunch of other areas around the state. I hear the same thing from out of state hunters Evey year " how the hell do you get close enough to get a good shot at these things"? Several years ago a my cousin had some friends come out from Pennsylvania the had permits for area 38 I found them some private land to hunt and let them go as they where all experienced hunters and I had to work. I figured it would be no problem, they all had what I figured where long range capable rifles, they shot and practiced 3 or 4 times a week. To make a long story short after 3 days of hunting hard they had exactly ZIP. I couldn't figure what they where doing wrong, on Saturday I took them out and got a great education on such things as judging distance, stealth, and the big one "Buck Fever". It took most of the day to get them there 4 goats it wasn't there fault they where just using back east hunting tactics out here and it didn't work. So the first thing I would recommend would be find an area that offers some cover, the central and western part of the state have areas of Sage and hills and generally make it easier to hunt. I'm not saying that you cant get a good shot at a goat out on the flats I do it every year I just cant explain a lifetime of experience in a paragraph. As far as finding a place to stay almost every little town will have a motel and a Café, maybe not 5 star but usually a nice place to stay. You will probably find it to be allot more friendly than some big motel in a city. Also driving 100 miles out here is nothing I drive 150 miles every year to where I hunt we rarely stay over night, so yes 300+ miles in one day is no big deal.
So find an area, if you don't draw find another, do some research. In resent years I found a cool little tool to look at possible areas, Google Earth. Yup bring up a satellite view of the areas your looking at and it will give you an idea of whats there besides roads and towns like you see on a map. You can call the game warden in the area you draw and he will be able to give you some ideas of access either public or private. If its private land call ahead try and get permission before you come out. Make reservations motels get a little busy that time of year. The most important thing Respect the Land, and the game. Years ago it was quite common to hear about "chasing goats" that is not acceptable at all any more. You will have much better luck if you spot from the truck and stalk on foot. I hope this helps, if I can help out when you decide where you want to hunt let me know.
Bob