Budapest, the capital of Hungary, is split by the Danube River into Buda (on the west) and Pest. The famous Chain Bridge connects both sides together. I recommend lodging near the Chain Bridge, as this will be convenient for visiting both sides and there seemed to be more dining and entertainment options nearby. Buda is the hilly old town, while Pest is flat and more populated. With the Hungarian Forint as the official currency, you'll find favorable pricing if you are converting Euros or USD.
Buda's elevation offers great views of the city. As you cross the Chain Bridge, opened in 1849, your first stop should be Buda Castle. A funicular is available if you don't want to walk up the hill. The castle was first erected in 1265 and has housed many Hungarian Kings. But like most castles from that time period, it has been rebuilt several times over the years due to wars and fires. Today, Buda Castle houses many museums, including the Hugarian National Gallery and Budapest History Museum.
Nearby is the Fisherman's Bastion with its famed spires and turrets. Opened in 2910, it is a great spot for photos. There is a hidden chapel under the ramparts that dates back to medeival times. Right behind the Bastion is the Matthias Church. The original building was destroyed by Mongols in 1241, so the version you are looking at today is from the 14th century and has had heavy restoration since. The impressive interior is worth a visit. I enjoyed wandering around this area.
At night, if you want a scene that really sets Budapest apart, head to a ruin bar. These are bars built in abandoned buildings, many without ceilings, that were left to rot after WWII. They don't look like much at all from the outside, but once you enter the scene is an incredible assault on your senses with lights, crazy decor, loud music, busy dance floors, and drinks galore. My favorites are Szimpla Kert (the original) and Fogashaz.
Start your day walking south on the Pest side to the Hungarian National Museum. It features displays on local history, art, and archaeology. A short walk away toward the Danube is the Central Market Hall. This is the largest and oldest indoor market in Budapest. It's packed with meats and produce, candies and pastries, and drinks aplenty. The second floor has many eateries and souvenir shops. I could've easily gone here every day! Then, take public transportation to see the beautiful Hungarian State Opera House from 1873. But don't just stare at the beautiful fascade from the outside - they offer a tour and mini-concert in the afternoon. Then, walk about a mile to the Danube shore to see the Parliament Building up close. It's a true architectural gem. This Gothic Revivial style building opened in 1904 and has 691 rooms. Be sure to take an evening dinner cruise of the Danube River to fully appreciate the Parliament Building when lit up beautifully.
Head to the nearest metro station to explore more great sites in the City Park. Heroes' Square is large and noted for its immense statues depicting chieftains, other important Hungarian leaders, and the Memorial Stone of Heroes. Surrounding the square is the Museum of Fine Arts and Palace of Art. Also in the park is Vajdahunyad Castle. It was built in 1896 as a celebration and features copies of several parts of iconic Hungarian buildings. Taking a stroll through this area is pleasant. Finally, visit the Széchenyi Thermal Bath, which is the largest medicial bath in Europe and supplied by two thermal springs. It now features 21 pools, saunas, and aquafitness. Lockers or cabins are available with your ticket. Even though it was crowded, I still found it to be relaxing. When it comes to eating, the famous New York Cafe is worth a visit. It is pricy and reservations are recommended, but there's a reason it's called the most beautiful cafe in the world. The food was of good quality. Then, head to another ruin bar to end the night.