Ryanair also fly from Dublin via London-Stansted to Leipzig-Altenburg, which is close by. Ryanair should be cheaper, but with fewer connections. Check on www.ryanair.com as they don't sell through travel agents. A train from Frankfurt takes 8 hours to get to Dresden. You can drive in about 5 hours if you want to chance the no-speed-limit autobahn experience.
Another option might be to land in Prague in the Czech Republic with Czech Air, who fly direct from Dublin. From there, the train to Dresden is about 4 hours, and Prague itself is a beautiful city. Whichever route you take, book early as summer flights will fill fast. US flights options will probably lead you to Frankfurt, then on to Dresden.
