Have a great trip. I have never been to Budapest, but if I did I would go to the Gellert Baths.

I like Vienna, and if you go you should go to the Belvedere and see the Klimts because they are not kitsch, they are astounding. The countryside round there isn't much to write home about, though the Wachau is pretty; you need to go further west in Austria for decent mountains. The Salzkammergut is rather 'Sound of Music'-y, but delightful and there's quite a lot to see there.