First of all, if you live in Seattle and haven't been to Discovery Park, go now. No excuses.
Now, this trip report is mainly to make a few comments about the park and trails, because even a seasoned visitor forgets and makes mistakes sometimes. See below.
We dropped by in the evening to go for a trail run, waiting for the heat to dissipate from the unseasonally warm weather. I personally like to park in the south lot off of Emerson - there's a honey bucket there for your use if needed (but be warned that the door doesn't latch well and it's well used) and easy access to the Loop Trail. For a short loop, stick to the Loop Trail, crossing streets and other trails and through the tunnel under Discovery Park Blvd back to the parking lot. This is about 2.9 miles or so, but don't be fooled by the short distance - you'll start by descending about 150 feet and then spend the rest of the run re-gaining the elevation through many rolling hills in the park. The trail is in good condition, and there are the occasional park benches along the way in case you want to sit and enjoy the view. You'll also have solitude if it's a cloudy weekday or evening.
If you are up for something a bit longer and steeper, do the outer loop. This begins as before on the Loop Trail, but then a left takes you on the South Beach trail, descending through the forest with occasional glimpses of the water and mountain to your left, down to the beach. Run along the beach for a bit and then explore the lighthouse, sticking to the outside (careful on the rocks!) of the point before joining up with the North Beach trail. From here you can see Golden Gardens and the marina just to the north, before the trail heads inland and you quickly gain back all that elevation in a series of unforgiving stairs. Try to run up all of them! I can't do it yet. When you hit the Loop Trail, follow it back to the parking lot for about 4.1 mile loop. This one is my favorite, as you can't beat the views along the water, and you certainly can't beat a sunset down by the lighthouse.
As I mentioned earlier, we waited until later in the evening to run so that we could escape the heat. The coastal breeze and fading sun did the job, but don't forget that the Loop Trail is in the woods. Despite the nearly full moon, we had little light on the trail. Bring a headlamp if you'll be there at sunset/dusk.
Also, you can't park down by the lighthouse. Yes, this means you have to walk a bit to see it (or take the shuttle on the weekends, but why do that when you can explore the park on your way?). You might get away with parking your car in front of all the No Parking signs for about 5 minutes, but almost every time I'm down there I see 1-2 cars get towed away.