Seattle gets really beautiful in spring. You will see the pink and white cherry blossoms everywhere. Depending on when the weather starts to get warmers, Cherry blossoms show up sometime in March and go through April 2nd to 3rd week. There will be many places you can see them, but one most popular spot I go to every year is the UW quad. This place is so popular that UW even maintains a separate twitter handle (@uwcherryblossom) to post peak reports etc.
If this is your first year, just do visit this place at least once. It does get crowded in the weekends so may be going early or a little late in the evening might help manage some crowds. This year, I plan to explore some new places to see cherry blossoms, so if I do end up finding a new secret place, I’ll share on this page soon.