There's plenty to see and do on the Pest side of the city, whether it's sightseeing, dining, going out or relaxation.
Sightseeing wise you definitely need to see the St-Stephen's Basilica and it's square. Though not as spectacular inside as some of most famous European cathedrals, partly because you can only enter through the side hall, so you can't see the whole cathedral. I saw on the pictures that the altar part looks amazing, but was not able to see anything from the side of the basilica. There's an option to climb up the stairs to see the panorama around, but we chose not to. I'm still traumatized by the Sagrada Familia experience I had, because I'm scared of heights. It was also the hottest day of our stay, so I was not particularly up to the task.