The best way to get to Dingle is, unfortunately for you, to drive there. I think the alternative is to take a train from Dublin to Tralee (most of those trains require one change), and then a bus from Tralee to Dingle (or a much more expensive cab ride). The train from Dublin to Tralee is about four hours. The bus from Tralee to Dingle is about 70 minutes. Time-wise, that is about the same as driving. It's about 60 euros per person, although maybe you could find a better price on a train.
You could fly Dublin to Kerry (not sure on cost). But then you'd have to take a bus to Tralee from there, and then the aforementioned bus from Tralee to Dingle. Or do a pretty expensive cab ride from the Kerry airport to Dingle.
So it is possible to do it with public transportation, but I think a car is much easier. Plus you would have the flexibility to make a stop or two along the way (there are plenty of things worth seeing in the middle of the island, which would also break up your drive).
No I don't think it'll be too rainy. We were in Dingle in mid-October a few years ago and the weather was pretty good overall. You basically have to plan for rain pretty much all the time, but it won't be any better or worse. You may have a reduced selection of pubs with live music at night, but there will still be a few.
If you are looking at somewhere else to go for a night or two, consider Kilkenny. You could get there by train from Dublin, and there is plenty to do there (old medieval town, a castle, a cathedral with a cool round tower, Smithwick's, etc.).
Have you already booked your flights? Could you set it so that you have Dingle and Shannon on either end to save yourself some time?
