These are some of the most common gathering places and recommended-in-tourist-books locations. How did you not know about them when you moved to the city?
Skip the puny crowded Needle and go to the tops of some of the tallest buildings downtown. Cheaper and with better views. I think their observation decks are around $3 to $5.
Carkeek park, the abandoned expressway ramp near the Aboretum for a little post-apocalyptic inspiration, for UW students (and let's face it most new students in Seattle are coming to UW) the WAC has cheaper boat rentals than Agua Verde, the Frye art museum (not so hidden, but always free). How about you add some of your own #2?
Go to Gasworks Park. It's free, there are lots of opportunities for photos, frisbee, picnics, and kites, and you can walk (or longboard) there from UW. Just DO NOT GO IN THE WATER.