Seattle's public transportation system is getting better every year. As Seattle's traffic gets steadily more congested, you might want to ditch the rental car and head for the light rail. Seattle's infamous double-decked Alaskan Way Viaduct, on State HWY 99, affords some of the best views of the waterfront and Puget Sound. It opened in 1954 but is scheduled for demolition so if you get here before then make sure to drive it for an entirely unique perspective.
If you decide to go the cycling route, make sure to stick to areas with well-marked bike lanes; Seattle drivers can be less than accommodating to cyclists. If you don't have your own two wheels, try Seattle's public bike share, Nice Ride.
Seattle doesn't have a subway but is in the process of building a light rail system that is mostly above ground but does take some deep dives to avoid heavy traffic areas. Currently the light rail is a great way to get from SEA-TAC airport to south-end destinations like Rainier Valley, Colombia City, and Mt. Baker. It ends at the downtown core, pretty close to Pike Place Market. There is a tiny section of Monorail called the South Lake Union Trolley that pretty much just serves Seattle Center attractions and the Amazon settlement of South Lake Union.
Buses are cheap ($2.50 fare) and will come in a timely manner, but it takes quite a long time and requires one or more transfers to get across the city. Make sure to bring exact change and be sure to ask for a "transfer" - these little slips of paper will get you on another bus if your route requires you to change buses.
If you find yourself with cash to burn, Seattle's rideshare services (Uber, Lyft, or Taxi), are without question the fastest way to get around. Car2Go also operates a robust fleet in Seattle so if you are a member, look for the blue and white smart cars dotted around the city and remember, parking is free with this fleet!