You can find samples of some of the older (and newer) Bluegrass songs on youtube. I consider that a good way of dabbling, so you know what you want to go buy

.
I was exposed to live Bluegrass regularly growing up, so I love it, and it's the music I dance to most naturally. (Not that I am a natural dancer in any sense of the word). My family also had some tapes of the older Bluegrass as well as a couple from newer bands. I suppose I prefer the older stuff to the new, but I'm not
that picky. I do prefer the upbeat stuff to the more slow/spiritual songs.
Yep, Bill Monroe is a good place to start

If you don't like his voice, his songs have been covered by plenty of other people.
Kentucky Waltz and
Blue Moon of Kentucky or
High Lonesome Sound. Or stick to his instrumentals:
Southern Flavor
You will also be familiar with
Down in the Valley to Pray since it was used in
O Brother... Doc Watson and Ricky Skaggs are old school, while Alison Krauss is a relative newcomer.
Here she is with Union Station.
The quintessential Bluegrass song is
Rocky Top Tennessee by the Osborne Brothers. But I suspect you won't like their voices. The song has been recorded by many other artists, so just hunt around and find a version you like. Personally, I think the instrumentation on this one can't be beat. My sister danced with our father to this song at her wedding. If I ever get married, I'm sure I'd do the same. We both remember dancing with him to Bluegrass growing up.
Osborne Brothers playing banjo but not singing (mostly 'Cripple Creek') I like the energy of songs like
Midnight Flyer, but
Don't Let the Smokey Mountain Smoke Get In Your Eyes might be what you are looking for.
Lester Flatt and Earl Scruggs are other big names. Here's Scruggs playing banjo on
Foggy Mountain Breakdown with Sonny Osbourne (No doubt the source of the name 'Soggy Bottom Boys' in
O Brother...was the Foggy Mountain Boys) But they really were a great duo.
Mountain Dew at Carnegie.
Wreck of Old '97 on
Beverly Hillbillies.
Salty Dog at the Grand Ole Opery.
Cripple Creek [This song does have lyrics, btw. They even sang it on
Sesame Street] Sorry the quality on these old clips isn't always great.
While Rocky Top and
Dueling Banjos are probably the best-known Bluegrass songs, my sister and I would always request
Fox on the Run I think we liked the harmony. Another clip from the Country Gentlemen is
Dark as a Dungeon - a bit more sobering than the other songs I've linked, so not my favorite.
Night Train to Memphis is good; at least, I like it better than
Nine Pound Hammer.
But I can throw in another non-Bluegrass song that reminds me of Bluegrass:
Seven Bridges Road by the Eagles. It's Southern Rock, but has some of the same spirit, to me. Of course, Jim and Jesse did Chuck Berry's
Johnny B. Goode and The Monkee's
Last Train to Clarksville, so I guess it's okay. The Osborne Brothers did John Denver's
Country Roads.
Nickel Creek started out Bluegrass, but I don't know that you'd still call them that.