Space Needle (https://www.spaceneedle.com/) Go on a clear day to see Rainier in all her glory. It is a touristic thing, but it is actually cool!
Seattle Center monorail: links downtown’s Westlake station to the area where the Space needle is. It’s a fun short ride.
Kerry Park: it has beautiful views of Seattle and Rainier.
Museums: Museum of pop culture (MoPOP), the Pacific Science Center (more for kids), the Chihuly Garden and Glass (not super cheap), the Seattle Art Museum, the Museum of Flight (for those interested, it is HUGE and worth a visit).
The Seattle Great Wheel is at the waterfront and has beautiful views of the city and nature around. Even if you don’t want to go in the great wheel, the walk from Pike Place to the waterfront, where the Great Wheel is, is very nice.
The Seattle Central Public Library’s architecture is quite cool.
The Pike Place Market has a ton of restaurants and store options, along with beautiful views of the bay. The first Starbucks store is here, though be warned that this not as interesting as it might sound. The gum wall is also here (https://www.pikeplacemarket.org/vendor/the-gum-wall/ ).
Gasworks park (https://en.wikipedia.org/wiki/Gas_Works_Park): bring a picnic or just enjoy the views.
Lake Union: enjoy the views of the Space Needle, rent a kayak or paddle board, swim/paddle with sea planes.
The Fremont Troll is a Seattle tradition (https://en.wikipedia.org/wiki/Fremont_Troll)
You might be interested in checking the Northwest Trolls (not related to the Fremont troll) that are in Seattle: https://www.nwtrolls.org/
Washington Park Arboretum has many nice walking trails just across the Montlake bridge from the university area. Also includes a Japanese garden. https://botanicgardens.uw.edu/washington-park-arboretum/
The Ballard Locks are a popular Seattle tourist destination. Throughout the summer, on weekends, they host concerts. Most importantly, people visit to see the fish ladder viewing room. In late July, you will probably see Chinook and/or Sockeye Salmon making their way through the fish ladder. https://ballardlocks.org/index.html
Not too far away from the locks is Discovery Park. Discovery Park is a 534-acre natural area park that has some nice trails, beautiful views of the Puget Sound and Olympic Mountain Range, and a lighthouse. https://www.seattle.gov/parks/allparks/discovery-park