Revised 9/22/2021
The Big Tree Ridge Trail offers a great hiking route to Cougar Mountain, and it's quite bus friendly.
From Seattle, ride Sound Transit 554 to Issaquah Transit Center.
Starting from Issaquah Transit Center, walk Newport Way westbound (good sidewalk on north side of road) about a half mile, until you see a white steel gate on the south side of Newport Way. There may be a more developed trailhead and parking lot here, as of 2021.
The new Big Tree Ridge Trail starts on the other side of the gate. At first the trail climbs gradually through woodlands, then it commences a steeper switchbacking ascent of the northeast slope of Cougar Mountain. The newly constructed trail is well built and graded, a joy to hike on. After a gain of 600', the trail tops out on a forested ridgeline with a bit of a view -log benches offer a place to sit and relax. Here the climb eases and the trail soon arrives at a junction with various other Cougar Mountain trails. If you go left, you will be on the Military Road Trail, a remnant of a very old pioneer route, which descends down to the Talus development. A right turn takes you onto the Red Cedar Trail, which ascends further up the mountain and ties in to the extensive Cougar Mountain trail system.
Warning: trail signage is not good in this area of Cougar Mountain and you will need a good map of Cougar Mountain Park, such as the "Green Trails Cougar Mountain" map, to find your way around on the extensive trail system here. The modern navigation option is to load a backcountry navigation app, such as Gaia GPS, on your smartphone. On-line maps now show most trails in the park. Even good old Google Maps shows trails, too. Of course, it is still good to have a paper map in case something bad happens to your phone.
There are a number of options. The simplest route is to hike back to Issaquah Transit Center and return to Seattle on the Sound Transit 554.
For a longer hike, cross Cougar Mountain and catch buses in the town of Newcastle. This route is long (6 miles) and complicated; you will need a good map of Cougar Mountain Park to do this. Exit Cougar Mountain Park using the new Cross Town Trail, where bus stops are located on Coal Creek Parkway.