The Loop Trail in Discovery Park offers a nice respite from the city, with views of the Puget Sound, open fields and forest. Given it's a loop, there are a couple of different options to begin the hike. The most common is to park at the East Visitor Center Parking lot and take a short connector trail up to the junction of the Loop Trail. You can hike the Loop Trail in either direction.
There are numerous trails criss-crossing through Discovery Park, many of which cross the Loop Trail. There is great signage along the trail to keep you on track, and cell phone reception is good as well, if you need to consult an online map. Enjoy the views of the Sound, and consider taking one of the side trails down to the Beach.