Most music in the western world came from western Europe. It's primarily based on secular dance forms such as reels, hornpipes and polkas (jigs too, but they are limited in some regions)
- the melodies consist of 4 bar phrases and are either 16 bars long or 32 bars long
- the chords used follow the I, IV, V pattern (if you're in the key of G then the chords will be G, C, D), sometimes with the relative minor (Em).
- the first part of the song (usually called the A part) consists of either 2 4 bar phrases repeated (8 bars) or 4 4 bar phrases repeated (16 bars). the 2nd part will be a duplicate of the first.
Understanding this structure makes it easy for you to parse out the different sections. Vocals will generally consist of a verse and a chorus, the chorus usually repeated after each verse with breaks in between. The breaks in bluegrass and a lot of folk styles may be improvised, the player making up melodies (consisting of variations on the arpeggios) over each chord.
The best way to learn how to play this kind of music is to get a copy of the Fiddler's Fakebook, which has 500 classic fiddle tunes in it. Learn as many as you can. Memorize them. Learn the chords and the melodies. Once you have about 50 tunes memorized (someone who's classically trained should be able to do it easily), start going around to jam sessions and Irish sessions in your area. Listen, write down the names of tunes and go home to learn them on your own time. Observe how the tunes are being played in the group. Once you think you have a handle on how the tunes should be played, ask the group leader if you can join.
Another way to do it is to find out where the contradances are in your area and start dancing. Bring your fiddle and ask if you can join the band for a couple tunes.
Finally, join your local bluegrass fan club - there are a bunch on facebook, or you can google it and find out where the closest one is for you. Go to the jam sessions, have fun!