The initial trail is well traveled and chances are you'll see many other hikers and runners on the loop.
What's great about this trail is that it's not that hard to get to and it is a huge park in Seattle. A little over a .5 miles and there is a great view of Elliott Bay - at this point you can chose to just continue along the Loop Trail
, and it'll only take about an hour. If you want to add a little more difficulty definitely go down to the lighthouse.
There is parking at the Discovery Park Educational Center. Start on the Loop Trail
and then look for the sign that indicates the trail down to the lighthouse. This is a fun run with some great views. It's only about 3.5 miles, but you can get a nice workout in.