Five days is definitely a good amount of time to explore the city and the surrounding areas! Below I'll sketch out a 5 day itinerary, but none of the days should be in any specific order. Feel free to rearrange based on weather, etc.
Day 1: Parliament, Shoe Memorial, Chain Bridge, Buda Castle, Fisherman's Bastion, and St. Mathias Cathedral.
- I'd recommend starting your morning off by visiting Parliament. Get tickets right away or in advance because they do sell out. Then walk alone the Danube to the Chain Bridge and visit the holocaust shoe memorial on the way.
- Buda Castle you can also either walk up via the royal gardens (there's an escalator that takes you up part of the way) or through fisherman's bastion or take public transportation or the funicular. I'd recommend going up through the gardens if you're able. You get great views of the city from the castle and there are a lot of look outs on either side.
- After exploring Buda Castle make your way over to Fisherman's Bastion and St. Mathias Cathedral. Stop for some lunch or at a cafe on your way over or browse some of the shops!
Day 2: Citadella/Gellert Hill, Market Hall, Dohany Street Synagogue, and the Jewish District.
- I recommend going up Gellert Hill/Citadella in the morning because it can get quite warm. You can either walk up or take public transportation or taxi. If you're able the walk is really nice and has a lot of good vantage points.
- Make your way down and across the Liberty Bridge to the Great Market Hall. Explore the food stalls and head upstairs for traditional Hungarian souvenirs and street food (try the Langos!).
- Walk or take the tram to Dohany Street Synagogue and then spend the rest of your day exploring and wandering the Jewish district. Check out Szimpla Kert during the day and maybe stop for dinner at Mazel Tov to end your day.
Day 3: St. Istvan's Basilica, Opera House, Andrassy Avenue, Heroes Square, and City Park.
- Start your day off at St. Istvan's Basilica (you get a great view of the city if you walk all the way up). Walk up Andrassy and then take a tour of the Opera House (some tours come with a mini concert!). Either walk the rest of the way or take the M1 to Heroes Square. Then explore city park and either walk back down Andrassy or take the M1.
Day 4: Day trip!
- Szendentre: Great little artist town to wander around, shop, eat, and explore! If you're feeling mighty adventurous make your way to Obuda (explore the cute square) and find your way to Berguson Bike Rental. The bike rental is super cheap and the ride is about 36km round trip. Just make sure to check the website to have have cash for their fully refundable deposit.
- Eger: Another great little town with a lot to explore. Walk up to the old fortress, get a glimpse of the minaret, walk the square, etc. Eger is also in a wine region so there's a bunch of wineries in the area you could tour on the way.
- Esztergom: Hungary's old capitol. The city sits along the Danube and is a great day trip.
Day 5: Bath day! And Margaret Island.
- There's tons to do on Margaret Island. Rent a vehicle/bike and tool around the island. Visit the ruins, climb the water tower, see the musical fountain, explore the gardens, etc.
- There's tons of baths so I will keep it brief and you can choose, which fits you best. My personal favorite is Kiraly, which is near Margaret Island. It's got a very traditional Turkish feel and is small and not very touristy. Definitely a local place. Szechenyi is the biggest and most popular with a massive combination of indoor and outdoor baths, but can be very busy and touristy. Gellert is in a nice hotel and has a great selection of indoor baths, a wave pool, outdoor pools, and a Swedish sauna. Rudas has a modern bath complex with a rooftop pool and traditional Turkish baths (watch what day you go because they have male/female only days).
Hope this has helped you!