There are three main trails that connect or pass through West Seattle, with all three converging at the west end of the lower West Seattle Bridge. One connects West Seattle to Alaskan Way and downtown - a popular commuting route, while another heads south along the Duwamish River and thru South Park. Both of these are through industrial areas and not particularly scenic. However, the third trail along Alki Beach is one of the highlights of the city, with beautiful views of downtown and the beach plus plenty of bars and restaurants.
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West Seattle Bridge Trail
This portion of the trail is a little over a mile. There's not much to see, it's mainly an industrial area, and there's usually some kind of construction going on. As long as you stay on the paved trail you will be fine. The map:
Starting from the Seattle side, follow it along the west side of the road, being very careful when crossing the truck entrance to the port, make eye contact first. To get to the portion of the trail that crosses the lower bridge, follow the signs to a U-turn route behind the bridge so you don't have to ever cross traffic. On the West Seattle side of the bridge you will come to a three-way junction, clearly signed:
From this point you can continue onto the Duwamish Trail or the Alki Beach Trail. Here's a video ride-thru as of summer 2012.
Alki Beach Trail
This 4-mile trail is the highlight of the West Seattle trails, some of the nicest views in the area. Because of this, you can expect heavy crowds on nice days. Luckily there is a separate 'wheels' lane so things generally flow well.
If you don't want to ride to the trail, it's easy to drive and park at the east end of the trail. The trail can serve as a starting point for a loop ride; there are many low-volume roads marked with sharrows to complete the rest of the route.
The less scenic and more utilitarian Duwamish Trail is a collection of paved trail, sidewalks, road riding, and train track hopping. It starts out from the bridge by paralleling West Marginal.
Follow the signs until you come to the stoplight, at which point you can cross to the east side and the trail proper.
From here, it passes through Herring's House Park.
The trail continues along West Marginal and then east along Highland Park and under 99. A recent extension to the trail follows Portland St, ending at Kenyon and 8th, both low-volume roads that will get you to the South Park Bridge. From the bridge, you can also continue to the south via 14th and then south on lightly-traveled West Marginal Pl which connects to the Green River Trail.