The Burke-Gilman trail and Sammamish trail are good places to start. Nice and flat with lots of parks and coffee stops and bike shops and such along the way. (and the RedHook brewery, and Hales brewery, and such) Riding on the trails might be a good way to get to know your new bike without having to deal with traffic and hills at the same time.
What part of Seattle are you in? There are lots of TE'ers in greater Seattle.
"If Americans want to live the American Dream, they should go to Denmark." - Richard Wilkinson