More recently, man made vantage points have given a new look at the region. The most well known is, obviously, the Space Needle and the views from there are, indeed, spectacular. However, there are other options like the Smith Tower and, in particular, the newly remodeled Sky View Observatory in the Columbia Center to consider.
This observation deck provides the highest public observation deck on the west coast (and west of the Mississippi) and the new enhancements make it a much more tourist friendly visit than before. It now offers a full 360 degree view around Seattle and there is more information for people who want to know what they are seeing. The down side is that prices went up: Regular tickets are now $12.50 a person ($9 for students, military, and seniors).
Not feeling like spending the money, but you are in the area anyways? The Columbia Tower also has what is known as "skybucks": A Starbucks located on the skylobby level of the building (floor 40). The view is not the same, but it is the cheapest way to enjoy a view from above inside of the downtown core of buildings.