Yeah Chainbangerz is the shizz. If you choose to play White River you can also check out Mando's but Chainbangerz is way better IMHO. Just call ahead to make sure they aren't randomly closed or going on lunch, which they seem to do every single time I go there. Not complaining because I love the shop, but just warning you.
-If you're a social person and like tight courses you might check out Mineral Springs. It's so small that you can't possibly play there without talking to someone.
-White River is in the summer layout right now, so you've got a good shot at getting lost there and only playing 15 holes.
-Lakewood is great and the one I would probably recommend. There's also a pro shop there that is run by Chainbangerz. Just expect a 30% markup because the county takes that much due to the shop being on park property.
-SeaTac will test your game to no end. Take a map or a picture of the map by hole one if you go there. That's my regular course and there are quite a few holes there that I haven't parred yet because it's so much beefier than I am.
-If you're heading north, try Terrace Creek. Make sure you PM someone who knows it though (me) before you go so you don't get lost on the back nine.
-Kitsap County has some incredible courses too, but if you're going to be downtown and only have time for one course? You won't have time to get over to NAD, Bud Pell, Fredericksen, or Fairgrounds. Come back sometime and check out Kitsap though. You won't regret it.