A lot of old-timey tunes may spring from folk sources that pre-date music theory, but there are in fact notes, chords and scales behind traditional numbers that conform to theory. Knowing some basic music theory will help you better understand and play these styles.
The first thing to be aware of is what key a tune is in. Most traditional, Irish and bluegrass tunes are fiddle tunes. The 4 strings of the fiddle are normally tuned G,D,A,E - in 5ths from thickest to lightest string. The majority of fiddle tunes are in keys that are open strings on the fiddle. I'm talking about the keys of G, D and A (rarely E). I'd say that 90% of the most-played tunes are in one of those 3 keys, so that will help you hone in.
Most songs/tunes start with a chord that has the same name as they key it's in. In other words, a song that starts with a G chord is probably in the key of G. An even better way to tell the key is to see what the last chord of the chorus or B-part is. That chord will almost always match the key. (This works the other way too - if someone tells you that a tune is in they key of G, you can safely assume that it probably begins with a G chord, and definitely "ends" with a G chord). Once you know the key you can make some educated guesses as to which chords to use.
The average traditional tune is going to have no more than 3 chords. Most likely the I, IV and V chords. If there are any other chords, these will likely be either the II chord or, if it's minor sounding, the VI chord. In the key of G, for example, the I, II, IV, V and VI chords are G, A, C, D and E-minor (Em). Sometimes the V chord is played as a 7th (D7 in the key of G), but that is completely up to you.
As far as knowing when to change chords, I can't really help you there. The people that do that well just kinda feel it and it's hard for them to convey how they know that. They just know that it's going to happen. (A guitar player might not know why he has to go from the I to the IV, he just knows that he has to go to the IV chord next). I'm still waiting for that ability! The good news is the chords to traditional tunes are rarely carved in stone. With the right fundamentals you can make educated guesses that should get you in the ballpark, and even if you are "wrong", as long as you are playing chords that are appropriate for that key, you won't sound too far off. You'll be harmonizing!