If you haven't already, check out 5 Days in Budapest - Part 1 first.
Here's a look at how we spent days 3-6 in Budapest (I've added links where I think they will be helpful to others who may be looking to visit Budapest. Some of them are affiliate links):
Day 4 - While the kids are away, the adults will play, haha. mrC and I purchased some Hop-On Hop-Off (HOHO) tickets and our first use was the boat tour that cruised right past our hotel. We jumped off at Margaret Island and spent an hour exploring until the next boat arrived. There was much to see and on a better weather day, we could have spent a lot more time there. We did find several fun things: the perfect spot for tourists to take photos, a very nice couple let us sit in their rental cart, lots of interesting restaurants, and the musical fountain. Here's a list of fun things to do that is worth a look.
TIP: Always carry change with you for the toilets. This is something as Americans (and expats in Hong Kong) that we were not used to. Even inside a shop you will probably have to pay for use.
We spent the afternoon exploring Castle Hill when youngestC finished her classes. We rode the HOHO bus (using our passes) almost to the top and then took a lift (had to pay) to the main grounds (if it sounds confusing, it was a little - but we got there ok). The place is huge and we did a lot of walking ---> loved the cobblestone but it's hard your feet after awhile. Doing our own self guided tour, here are the main parts that we saw: Royal Palace, Eugene of Savoy monument and Matthias Fountain, Sandor Palace (President lives here now), Matthias Church (we toured inside too), and the Fisherman's Bastion.
We took the funicular down from Castle Hill (we did not have great views - I recommend getting in the "bottom" car). Instead of waiting for the next HOHO bus, we walked across the Chain Bridge back to our favorite bistro for some dinner on the Danube. It was chilly, but they offer fleece blankets for people sitting outside. After dinner we used our HOHO passes for the night boat tour. This is a must-do if you are in Budapest. The city at night is stunning.
Day 5 - After a very busy day 4, our final day in Budapest was a bit of a rainout. mrC and I trekked through the rain to visit the beautiful St. Stephens Basilica. The ferris wheel was closed, but the views would have been amazing. We toured the inside of the basilica and even braved the pouring rain to visit the cupola. (side note: my Athleta rain coat was a lifesaver! I've had it for a few years, but here is a similar one.)
We took our workouts to the Marriott fitness center and were dully impressed with the facilities. Another good way to avoid the rain is to shop, so we visited Váci Street for some souvenirs and food.
Day 6 - Our final day was really just a morning. We took advantage of the good weather and another chance to run together. We took the tram to the Margaret Island bridge and ran to the island from there. On our run we got to see a little more of the island including a small zoo that seemed to feature deer and the ruins of a convent.
TIP: You probably know to travel with a converter, but this one is by far the best I have used. youngestC got hers at Target, but I ordered mine from Amazon.
What we missed (aka saving for next time? I like to be optimistic):
Thermal Baths - I was so sad not to get to do this since it is literally a must-do when visiting Budapest. The Rudas and Szechenyi thermal baths right in city center came highly recommended.
Ruin Bar - A popular way to have a drink is to visit a bar that is set up in the ruins of a parking garage, factory or any dilated building for that matter. We had hoped to have at least one drink at Szimpla Kert, known as the original