You can start and end at the visitor center, which has brochures and good info.
There are three trails. The first two are really just walks and are by far the most crowded. The first one, The Hall of Mosses Trail, is wheel-chair accessible and is 3/4 of a mile. The second "Spruce Nature Trail" is 1.25 miles.
The third, The Hoh River Trail, at about 18 miles long, goes all the way up into the Olympic Mountain glaciers. It's much less crowded so you can enjoy the quiet and the beauty of the forest, especially after the first mile. I like to hike this trail as far in as I want and then just turn around and hike back out.
If you get a permit from the Wilderness Information Center in Port Angeles, you can backpack and stay overnight up in the higher elevations of the Hoh River Trail, in the Olympic Mountains. This trail goes up about 18 miles, into glacier areas, and I've read that you can connect to other trails and hike down to the Sol Duc Hot Springs area in the north of the peninsula. I stick with day-hiking myself.
Fondest memory: Old growth trees seem to soar into the sky - it makes me dizzy just to look up at them sometimes. They never fail to fill me with wonder. Sitka Spruce, Western Hemlock, Douglas Fir, and Western Red Cedar populate the Hoh Rainforest, but my favorites are the big-leaf Maples. Moss carpets and drapes the trees, and ferns cover the forest floor.
Fondest memory: There's no "lodge" with a dining room, and no grocery stores, general stores, or restaurants for about 45 minutes around, so don't think you'll stock up on stuff when you get there. The nearest town is Forks (about 45 minutes away).