Helpful would be dates. Different things work at different times of the year. And a budget.
Eger and the Tokay region (Furmint, Hárslevelű, Sárgamuskotály) are a nice overnight trip from Budapest. But then another great overnight trip is Pecs and the wine region in that direction (Italian Riesling, Furmint, Chardonnay, Cirfandli, Hárslevelű, Muscat Ottonel, Királyleányka, Tramini and Rajnai Rizling). I would do Eger in warmer weather and Pecs in cooler weather. Then there is some wine production near Budapest its self that makes for a good day trip.
Balaton can be as long a trip as Eger so keep that in mind. For me the most interesting part of the lake is the artist village of Tihany. Great overnight trip.
Budapest its self is best served in no less than 3 full days, with a full day being a day you wake and go to bed in Budapest.
The must sees are going to be based on your interests. I don't know of many places that have better food if you are a foodie. This may be the pate capital of the world if you tasted tends in that direction. Then there is Empire history, World War History, Jewish History, Soviet History, amazing Architecture (Anthony Bourdaiin refers to the intensity of the architecture in Budapest as Architectural Porn), 19th century cultural history which includes spectacular coffee houses and bath houses; and the Music and Theater. The last topics of culture and music I think rival those any place in the world because of the quality and because that they are still very deeply entwined into the current culture and not just for tourist trade. Visually this town is hard to beat.
Between Budapest and Vienna consider a night in Gyor. Sleep in a converted monastery and visit the archabbey at Pannonhalma.
