Some great suggestions. This is such a wide-open category. Here are a few more suggestions that may help ease into country and bluegrass. Some you will find easier to acquire on ceedee.
Any of the Gerry Garcia, David Grisman collaborations. A.K.A, Grateful Dawg music.
Old an In the Way Garcia, Grisman, Vassar Clements, etc.
Flatt and Scruggs, yeah there is much more here than the theme to the Beverly Hillbillys but even that tune is a great example of their music.
Ricky Skaggs early stuff and his more recent stuff with his band Kentucky Thunder.
Another sub-genre that I enjoy from time to time is Texas swing. This, of course, means Bob Wills. I think a more palatable form of this music for the adventurous is the band Asleep At the Wheel.
If you're interested in country rock, I'd start with bands like The Byrds, the afore mentioned Burrito Brothers, Marshall Tucker, Charlie Daniels Band, early Poco, Pure Prairie League.
And it just goes on and on. We haven't really gotten into the hard-core country stuff, but that's a place where I tread lightly.