I have been out to Wyoming twice on DIY hunts. Both trips we only hunted public land.
In 2010, we hunted unit 7 near Newcastle. We drew this unit without any preference points, but since then it has gained popularity and you might need 1 point to draw. We went the first weekend of the season and saw a ton of goats, but also a ton of other hunters. There were two of us and we tagged out in two days.
Here's my antelope
This last year, we hunted in Unit 63. We tried to draw for Unit 9, but were unsuccessful. After the draw, all of the leftover tags will be for sale. These become available at a certain time/day and it is first come first serve. I got on as soon as it opened and got the tags. Unit 63 is considered somewhat of a trophy unit.
This time we were a little smarter. We waited until the 3rd week of the season to head out. We tagged out our 1st morning by noon and only ran into 1 other hunter.
My best advice is to become very familiar with the Wyoming game and fish website.
1st: figure out what units you will have a good chance of drawing in. This will probalby narrow it down to 10 or so units.
http://wgfd.wyo.gov/web2011/Departments/Hunting/pdfs/DRAW_ANT_RD_NONRES_20120002491.pdf
2nd: check for public land or get a landowners list. You can get names of landowners who allow hunting, for free or small trespass fee, from the regional offices found towards the end of the non-resident hunting regs.
http://gf.state.wy.us/plpwhmprogram/frmWalkinHuntingHome.aspx
3rd: fill out application and wait for the draw. If you don't draw, you will probably be able to get tags in the leftover draw. However, these areas usually don't have great public access.
Good luck and if you need any more info shoot me a PM.