Ach, I envy the both of you for your trip to Budapest! The city is best explored by just walking around and taking a random turn. There are many, many lovely spots. Not sightseeing spots, but just simple streets and buildings that have a big charm!

As there were just elections have a look out for demonstrations on the main places of the city. It might get pretty passionate.

A tip: Visit the Cafe Gerbeaud! it is a famous cafe and the dishes are addictive (especially the diplomat pudding!!) Eating at local restaurants can be a better treat than the more expensive touristy spots. Taking a walk down the Donau Promenade (especially at night) is also a treat. Visit the Margarete Island (it's pretty) and the one next to it (if you are up for celebrating).

Oh, basically just enjoy the city at your leisure. It is a wonderful city and there is enough to discover without having to rush anything.

Have fun!

PS: Köszönöm means thank you and jó napot kívánok means Good day. That should get you quite far :)