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Well, this trip came out at the last minute, and this could be interesting if you're thinking low season, but try to find accommodation in the middle of August a week before the trip... that was tricky!
I was considering either Prague or Budapest, and in the end I had to take Prague, which came out as a lovely pick. I spent a full busy week and I still missed a few sites to visit.
Prague is a very interesting city, with the biggest old quarter, and by the way, it is the original one, it hasn't been reconstructed, which is remarkable.
If you hire a car in Prague, the city will be more accessible to you, especially because taxicabs are way too expensive.