Really depends to some degree on where you are staying. Its a big city and the laid back time is more laid back if close to "home". Time of year will change things to.
If I were to assume you are coming in warm(er) weather and have chosen wisely and are stayed someplace along Andrassy ut (avenue) (the Paris Champs-Élysées was an attempt to copy Andrassy ut ... okay, not true, but you will see Budapest did it better), I would suggest you just walk up Andrassy ut and enjoy the architecture.
At the end of Andarssy ut Heroe's Square with some interesting iconography of the begining history of Hungary. Beyond that in City Park, still walking, you can find the Hungarian House of Music (just walk in and look around, most are not disapointed in the modern marvel), the Museum of Ethnography (just walk the roof top. again, most are not disapointed by the effort). Stop at the outside Zöld Küllő beer garden for a drink and maybe lunch? Walk thru the park to the Széchenyi Baths and take a long soak.
Heading back if the topic interests you there is the Zoo which besides being a pretty darn good zoo has some exceptional Art Nouveau architecture. Or the circuis which is now a really great performance theater.
Then if you are tired, the M1 metro runs directly under Andrassy ut right back to your hotel (if you chose wisely). But before bed time, be sure to walk past the Opera House to the opposite end of Andrassy ut and on to the Danube River and have a drink or dinner at DunaCorso and enjoy the lights of the castle. Few places more beautiful in the world. Again, the M1 will bring you home as it extends just short of all the way to the river.
Andrassy ut https://youtu.be/f6vc7KYPdEk?feature=shared
M1 Metro https://youtu.be/f2J5mKdbbf8?feature=shared
Heroe's Square https://youtu.be/VLrlnmL3_u4?feature=shared
House of Music https://youtu.be/yNzw6h3blQw?feature=shared
Museum of Ethnography https://youtu.be/Aey_g__dTrc?feature=shared
Zöld Küllő Beer Garden https://kep.cdn.indexvas.hu/welove-media/28/zold-kullo_hartyanyi-norbert_20170802.jpg
Széchenyi Baths https://youtu.be/a01uKNLK1eM?feature=shared
Opera House https://youtu.be/taoMYdBG5f8?feature=shared
DunaCorso https://www.dunacorso.hu/templ/gfx/content/galeria/3.jpg
Zoo https://youtu.be/D_hOsD0JSpw?feature=shared
Circus https://youtu.be/apz1izRgEZY?feature=shared
Now, if you arent staying along Andrassy ut or in warm(er) weather, there are other ideas.....