My brother in law married a girl from székesfehérvár and we visited for some long trips. Budapest had the most going on from a lot of ways, modern and historic, but székesfehérvár Is a neat older city not far from Budapest. There are lots of roman ruins scattered everywhere and loads of history. It's not a great time of year for it but my favorite was time spent around lake balaton. You should still go - lots of old Hungary was built around the lake.
Touring around the Danube was neat too, visegrad is badass and near to Budapest also