People aren't too forthcoming with "best" places, but "known" places that are consistent producers are a little easier to come by. Try the narrows (Tacoma side, you walk up from titlow) or Titlow park (nice inlet just down from the narrows) or cross the bridge and fish doc's beach.
Otherwise, it's up to you to explore. If you search the site, you'll find information on the kind of habitat which is ideal, although I've caught them over sand, large rip-rap, and cobble-type bottoms, in all tides, and I don't really get out for them that much. So the best spot is where the fish are.
From my experience lately, the SRC have been pretty well scattered on Puget Sound. The Fall and Spring are usually the best SRC fishing times on the Sound. In a couple of months they will be heading towards estuaries in hot pursuit of the chum and pink salmon. So don't give up on the SRC if you don't have much success now.
If you get lucky, you might be able to pick up a few SRC at a location before they move on. As Steve said fish early or late in the day when it is sunny. Low light and cloudy days are best for SRC and particularly silver salmon.
It is a challenging/addictive fisheries since every day is usually different and most trips you will learn/experience something new. That is the beauty of the Puget Sound fisheries along with being able enjoy God's created environment.
Learn as much as you can about this fisheries so that you will begin to have successful outings. Two good books: "Fly-Fishing Coastal Cutthroat Trout" by Les Johnson and "The Estuary Flyfisher" by Steve Raymond. Also do a search on this site. But better yet go out with someone who has some knowledge of this fisheries.
Hope that you have some successful fishing trips on Puget Sound in the near future and that you will enjoy this fisheries.
You said it well; perfectly in fact. It seems that many of our new coastal cutthroat anglers do not realize that we all have to search to find cutthroat every time out. There are no pat hands whether we've been fishing for 5 years or 50. This is indeed the joy of the sport in my view. I would hope that those new to the sport will eventually embrace the entire spectrum of fishing for coastal cutthroat -- a fragile and world-class game fish -- and grow beyond simply asking "Where are the SRCs?"
You have to be willing to get skunked if you fish SRC. Sometimes even the good spots hold no fish.
While searuns are quite readly caught from shore, having a boat enables you to move quite easily. The challenge is staying close enough to shore. One good rule is to make sure you can see the bottom. (of course in low light this can't always be done) Another good rule whether you are fishing from shore or boat is to practice stealth. These fish have all of puget Sound at their disposal (enough shoreline to stretch from here to S. California and back!) So why would they hang out in area where a fisherman is splashing the water or clunking around in the boat? Think like a hunter and stalk your prey...you will find more and bigger fish. These fish are some times in very shallow water, sometimes in 12" or less. If you are waist deep in the water you may have just scared away all the fish. I have caught SRC by casting to shore from a boat and my fly was landing 2' from the beach.
Buy Les' book, it'll not only make you a better SRC fisherman, it'll inspire you to go out more often.