A most wonderful city, the Pearl of the Danube! Buda and Pest, situated on the two banks, are great, one being mountainous with castles and old mansions, and the other being low with parks, restaurants and nightlife. It is a true spa-city, one of the main attractions being the thermal baths. Make sure you visit the Szecsenyi Baths: old beautiful building (newly renovated), healing water with lots of minerals and micro-elements and fountains/waves/jakuzzi for fun! Can be visited around the year as the water is warm/hot, so you won't freeze even in winter in the open-air pools. Bliss!
Gellert Baths (in Gellert hotel): probably the most expensive baths located in a fancy hotel on Buda side with a pool on the roof. It has separate sections for men and women for those who consider it fun :)
Underground: the yellow line ("Foldalatti") is over 100 y.o. and is the oldest on the continent! It is 11 stops long and runs through the centre of the city on Pest side. There are 2 more longer lines (looking exactly like trains in Moscow!), and a 4th is being constructed now.
Hungarian cuisine is famous for gulyas, paprika and halaszle (fish soup), which I recommend trying in small family-owned restaurants you will find all over the place.
And after a dinner, take a walk in the night city: take Szecsenyi Chain Bridge to cross the Danube from Pest to Buda or back And as you are walking at night, you can see many lights glittering in the sky - they are in Buda hills calling you up to enjoy the panorama of the night city. Go up the Gellert hill and enjoy the purple -pink colors of the city when it is waking up in the morning...
I spent 3 years there and keep coming back. I love the city and hope you will enjoy it, too.