I went to Uppsala last year. These are the sights I had on my list (most of which I didn't have time to see); I think they're listed roughly from north to south. The black triangles are Rick's prioritization:
Gamla Uppsala: Royal burial mounds (always open and free), good small museum with artifacts, 12C church (I think free) surrounded by evocative cemeteries, and tiny church museum (free?). 5 km N of town; Bus 2 (as of the time of my research) from Stora Torget, 0:15. www.raa.se/gamlauppsala
▲Linnetradgaarden, 27 Svartbacksgatan: 2.5 acres; very nice. Annuals right, perennials left. Gardens may be free after 5 PM. There's also a small house-museum, the Linnemuseet. www.linnaeus.uu.se
Upsala Konsert & Kongress, Vaksala Torg 1: Superb view of entire city from top floor (escalator).
Radhuset, Stora Torget 10: 18C former town hall, now elegant department store?
Riverfront has sidewalks on both sides and artworks that double as benches. Many beautiful houses from 17C on.
▲▲Uppsala Domkyrkan (1435), Domkyrkoplan: Sweden’s largest cathedral and perhaps most beautiful. Surrounding area also lovely. 15C with mainly 19C interior decor. Gorgeous Baroque pulpit and important side chapels. Treasury (Skattkammaren) has exceptional collection of medieval textiles, swords and crowns. There may be tours offered in English. Or see whether the gift shop is still selling a descriptive booklet. Church open daily and free; restricted access during services and concerts. Main service Sun 1100 and concerts Sat afternoon at the time of my visit. Skattkammaren (entry fee) has somewhat shorter hours; check the website. www.svenskakyrkan.se/uppsala/welcome-to-uppsala-cathedral
Slottsbiografen (Castle Cinema), Nedre Slottsgatan 6D: Opened in 1914 and restored to original Art Deco glory; shows art films.
▲▲Gustavianum, Akademigatan 3: University museum with mostly-scientific displays and archeological holdings. Augsburg Art cabinet has miniature works of art. Was closed at the time of my visit but supposed to reopen sometime this year. www.gustavianum.uu.se.
The university area is beautiful. I think I recall some hilliness.
▲Carolina Rediviva, Carolinabacken: Library. Exhibition room used to be free Monday-Friday and have a fee on Saturday. www.ub.uu.se.
Vasaborgen Uppsala Slott, Drottning Christinas Vag: Castle only viewable on tour but views and botanical garden are great. Don't know that it's open this early in the year. www.vasaborgen.se
Uppsala Konstmuseum, Castle Entrance E, Drottning Christinas vag 1E: Contemporary art.
Botaniska Tradgarden, Villavagen 6-8: Large university gardens. Used to be free except for tropical greenhouse. www.botan.uu.se/?languageId=1
Bror Hjorth Hus, Norbyvagen 26: Art of Bror Hjorth; rather naïve. Bus 2 or 7 from Stora Torget to Evolutionsmuseet or Birkagatan stops, then 300 m walk. Free as of 2022. www.brorhjorthshus.se
Hammarby: Linnaeus’s summer home, surrounded by lush gardens, 9 mi from Uppsala. Old-fashioned garden flowers, nature walk and cafe. Not sure how much is open in May. botan.uu.se/our-gardens/linnaeus-hammarby/visit-us/