The genre "country" music used to be a catch-all for rural folk music, so in that sense bluegrass is a product of "country" music. But in terms of how people talk about the two today, I wouldn't refer to bluegrass as country music. That's not incorrect but people tend to get Bill Monroe confused with Garth Brooks when you interchange the terms.
As far as recommendations go, I've got to mention Doc Watson, from my home county (Watauga) in NC. He plays bluegrass and a lot of other traditional folk, country, and blues.
Old-time music is excellent. Old shit from the Appalachian mountains. It was one of the major sources of inspiration for bluegrass and really does more for me than bluegrass ever has. Try the Hollow Rock String Band and the Fuzzy Mountain String Band.
KILLTULLY, would you repost a couple of those Hank III samples? They aren't working anymore. I'm probably gonna see him in Raleigh in about two weeks but I've only heard a song or two.