Thank you all for the reply. I'm still a newbie at this so am not familiar with the airlines abbreviation. The two carriers are indeed Aer Lingus and Austrian (Tyrolian) air. I'm glad that I can pick up my luggage.
During my research, I've found that the connection to Prague are somewhat difficult. I was trying to minimize connection as well as arriving at a good time. Vienna seems like the best for that. Zurich was another possible connection point, but that would mean I backtracked multiple times!