I strongly endorse Fred McCormack's choice of "Back in the Saddle Again." This is a terrific introductory compilation of Cowboy Songs.
Another one is "Vintage Country & Western" on Gallerie GALE450, distributed out of the UK by Demon Music. I checked the website for Demon, and it seems this may be out of print. It has a lot of cowboy, and some western swing and bluegrass. Highly recommended if you can find it.
My all time favorite cowboy music is (early) Sons of the Pioneers, 1930s and early 1940s, and Tex Ritter who was a collector of this music as well as singer and western movie star.
I also like the Sons of the San Jouquin, Brenn Hill, as well as the earlier noted Don Edwards (my current favorite) and Michael Marrtin Murphy. And definitely Ian Tyson.
For those who have internet access--and, duh, I guess that's all of us--I recommend two sites: Saddle Songs at live365.com; and Around the Campfire at classicheartland.com heard live at 7am cental time USA, and repeated several times during the week. Both will give a great introductory variety of Cowboy Songs, old and new.
Welcome to my world...this IS my favorite type of music.