Attempted a quick google maps search for a reputable archery shop near you but unfortunately there isn’t much near Port Angeles. There does seem to be a local archery range that goes by the name “Wapiti Bowmen” that may be worth checking out. I couldn’t find a web page otherwise I’d add a link. As a general rule, archers are kind people that love to share their experience with others who are interested. I suspect you could find a great mentor at that archery range if you get in touch with them or start shooting there.
As for Olympia and Portland, there seem to be many options, especially in Portland (10+ by my search). I have no personal experience in that area but do have a fair amount of archery experience. If it was me as a new archer, I’d choose a shop based on the reviews and the brands they carry. Also be sure they have a range on site so you can try shooting the bows you’re interested in. Just be aware that if you need service on your bow in the future, you may need to make the trip to them again or find another authorized dealer so I would buy as close to home as possible. I’ve added a few suggestions below to get you started…
This shop is probably your best option close to home. It has great reviews and stocks many brands previously mentioned including diamond, bowtech and matthews/mission:
There’s also an “Archery World” with great reviews near Olympia based on google maps. They are a national big box archery store and my experiences with them in the past have been positive. There are two archery world locations in Portland as well.
www.archeryworld.net
Archery is a fun hobby and an awesome way to make hunting both more challenging and more rewarding…good luck!
Edit - It seems I was mistaken, Archery World in that region seems to be just 4 stores in Washington and Oregon. I had a brain fart when I mentioned they were a national big box store and had mixed them up with a similar store name that existed all across the Midwest where I grew up. The fact they are a small store makes me want to recommend them even more.