Dane,
Seward park in south Seattle has a lot of nice trees. You can get a little off of the trail and avoid most people. There are some nice spreading Maples, and if you are interested some taller 150ft+ Douglas firs. Some wildlife and great views of lake Washington from the trees!
As for the best climbs in the state I don't know.
-Some really great Sitka Spruce and Doug Firs can be found on SR 20/ The North Cascades Highway. Try near Lake Diablo or Ross Lake. If you hike up and climb a tall tree on the hillside you can get a spectacular view of the lake.
-The peninsula has some huge, so I assume really fun to climb, cedars and doug firs. Just hike around Lake Quinalt in the Olympic Forest and look up. Haven't climbed there yet, but will this summer!
-I'm new to this and haven't paid my dues so if there is a "know" to be in, I'm not in it.
And now onto your "newbie" questions:
*With everything I'm about tell you: Don't consider this qualified instruction, I'm just giving trying to give you the gist of it.
Q: Once you're at the top of your rope, what can you do to continue?
A:
Note: Many people make the switch from ascender (Single Rope Technique or SRT)climbing to a doubled rope climbing technique when in the canopy. If I wanted to stay SRT this is how I would continue climbing.
*Key Items: 1. Throwbag and throwline (A bullet shaped bag filled with lead shot with a long cord attached. Think of a streamlined rock on a string.) 2. Short piece-10ft- of rope/webbing/flipline to use as safety-line/tie-in when changing anchors
First, you'll want both ends of the rope accessible. So instead of tying off to the base of the tree, try to tie off around just the branch itself using a sliding knot such as a running bowline (in this setup the force on the branch isn't multiplied by a factor of up to 2 as Treezybreez mentioned). Attach the free end of the rope to your saddle for later use. Then you can ascend to this anchor spot.
Second, you are at your anchor. Use your safety line (#2 in key items) and tie/cip yourself onto the same anchor branch you used to ascend. Now use your throwbag with throwline attached (key item #1): Tie the free end of the throwline to yourself (so you don't lose it) and toss the other end with throwbag over a higher crotch in the tree, and get the throwbag back to you. Untie the throwbag and tie the end of the throwline onto the end of the rope (it should be clipped to your saddle from step 1 otherwise, haul it up hand over hand from the ground). Pull on the other end of the throwline (should be tied to your saddle) to haul your rope over the branch.
Third, you should now have a loop of rope over a higher anchor point. You can untie your sliding knot from your first anchor, and use this end to tie the sliding knot over the higher branch. Tighten it up and now ascend to this new anchor. To go higher repeat steps two and three.
Descent, now you wanna come down. Tie your throwline onto the sliding part of your knot at the anchor (this will be used to take the rope down later). Now use your rappel device and slide down. If you have done multiple up pitches, make sure the bottom of your rope reaches the ground. If so, slide to the ground, then pull on your throwline to haul the sliding knot down and untie the rope. If it doesn't reach the ground you'll need to switch anchors.
To switch anchors on descent, descend to a sturdy branch and tie in using your safety line. Tie the free end of the rope to your saddle. Pull on your throwline to haul the sliding knot down and untie the rope. Retie a your rope with a sliding knot on this new lower anchor. To switch anchors again, just repeat the last two steps.
Q: How do you get up even higher into the tree, or move horizontally within it?
A: I just covered moving up in the previous section. Ascenders don't work so well for side to side movement. This is mainly done using doubled rope techniques. You set up two anchors, so you are hanging in a V between the ropes. You then ascend toward one anchor while loosening the other. It allows good horizontal movement. There are other ways to do this, but this is how I know how to do it.
Q: Is it possible to move from one tree to another?
A: Yes! This move is called a traverse. Short traverses can be done like anchor changes in doubled rope climbing, but they get more complicated as the distances increase.
Hope that helps! Again, this is just intended to give you an idea of how to climb and reset anchors using SRT -- It's not an instruction manual.