There are a couple of really good sites to get diddly music (mine being one of them of course.. well I would say that wouldn't I) and the Dublin Uni. site, which is also very good. I don't have the url for it to hand, but you can find a link from the Prof's pages, HERE
I admire anyone who can play the fiddle and as I've said before, given the magic three wishes, number one would be to wake up tomorrow and be able to play said instrument as good as, if not better than, Dave Swarbrick, who in my umble opinion is one of the all time greats (self taught as well).
I find it interesting when people ask for "beginner's pieces", as these vary tremendously depending upon the instrument being played (or so it seems) and more so if you are talking about the dancers... the session I play in tend to sneer at tunes like Rakes of Mallow and Off She Goes as being "beginner's tunes", as these are the ones the very young dancers call out... personally I think the latter of them is on eof the most demanding to play on the guitar.. so it's horses for courses..
the best advice that I can give you Don is to stick at it.. and I wish you all the best in your endeavours..