^^ That’s a great itinerary which will keep the OP busy for the duration without ever leaving Seattle.
nobody has mentioned the wonderful EMP museum yet. http://www.empmuseum.org/ located in the seattle center a must see after the Chihuly museum. Check out discount passes for the seattle center sites - they can save you some money - if you buy admission to 2-3 of them together. In the summer they stamp your hand when you visit the chihuly during the day and allow you to come back again that same night (the museum is lovely but different in the day vs. night.)
If you like the water, maybe a boat ride that include south lake union (they do seaplane tours there) and then visit the quaint museum of history and industry - a good introduction to the history of seattle and they have what looks to be a great toys exhibit coming starting in July. http://www.mohai.org/exhibits
I would skip pioneer square - nothing to see except homeless people - and when you do the international district hit the wing luke museum - http://www.wingluke.org/ - they have a cool exhibit on Bruce Lee
I adored the museum of flight -it is a long bus ride away http://www.museumofflight.org/
If you get a chance go to a Mariners game - the stadium is great and although large has a family/small town feel
@HiToWaMom, I am laughing at how CC edited out the key word on the statue. OP, it lives under a bridge, and is a legit alternate use of the word we can’t use out here. 
I was just in Seattle last month visiting our son, who lives in the Capitol Hill area and works downtown. We were just there for a long weekend and did the following: Pike Creek Market, Sky View Observatory (Excellent view of the skyline, cheaper than the needle, less crowded and higher!), took a ferry to Bainbridge Island, Did the Underground tour https://www.tripadvisor.com/Attraction_Review-g60878-d1089321-Reviews-Bill_Speidel_s_Underground_Tour-Seattle_Washington.html, Did a walk through of the Seattle public library and went to the top- 10 floors, Visited the Chihuly Garden and Glass (wonderful!), we did do the Space Needle but just so we could say we did. I didn’t think it was worth it compared to the observatory.
If you go to Chihuly Gardens, make sure to get there via the monorail (leaves from Macy’s downtown) and stop in at the Armory https://www.tripadvisor.com/ShowUserReviews-g60878-d143565-r178965751-Seattle_Center-Seattle_Washington.html They generally have something going on. food court and clean bathrooms. We didn’t do the EMP museum. Our son had been and the exhibits at that time didn’t interest us. However, we did go to this place https://www.tripadvisor.com/Attraction_Review-g60878-d141518-Reviews-Museum_of_History_Industry-Seattle_Washington.html and really enjoyed it. It’s right on Lake Union and you can take a sailboat ride right there too. We used public transportation and took buses and the light rail all over the city and didn’t feel the need to rent a car.
THE best place to eat is Din Tai Fung https://www.tripadvisor.com/Attraction_Review-g60878-d141518-Reviews-Museum_of_History_Industry-Seattle_Washington.html and it’s right next to the University of Washington.
We really liked everything about Seattle and our son loves it.
There are many lovely inns and B&B’s in the San Juans or Bainbridge/whidbey however without a car or bike (with a backpack) it will be quite challenging to get to. Uber/lyft is limited outside the city and our public transportation is really pathetic compared to most cities of comparable size.
I’d look for a B&B in Wallingford/Freemont/Ballard/queen Anne where you can walk to things and there is a good selection of food, music etc. far easier to get to ferries and do day trips,
Absolutely agree with the canoe through the arboretum. Rent canoes from the UW or kayaks at Aqua Verde (my vote) and then have a great lunch there afterward.
Also, Keens are the perfect waling shoe for here in the summer that double as a hiking shoe and quick dry in the rain. Completely agree, pack for 50-90 and bring a rain jacket, skip the umbrella.
Nothing says tourist like umbrella
We stayed at The Moore Hotel. Perfect location. Old, kinda funky place. You can get a deal of a room if you’re okay with a shared bathroom. We opted for a larger room with traditional bath and a mini-kitchen and included a coffeepot and hair dryer. From the reviews, that wasn’t true for all the rooms. No A/C but they have fans. https://www.tripadvisor.com/Hotel_Review-g60878-d100544-Reviews-Moore_Hotel-Seattle_Washington.html We just walked or took public transportation everywhere. However, there is a zip car place and a hertz rental within a block of the Moore Hotel. Traffic in Seattle looked pretty horrific.
i’ve visited seattle many, many times over the years. I second the arboretum tours via kayak (so cool); the ballard locks, and will mention how we liked the touristy boat tours that go by the shipping docks.
I’ll add something quirky and just cool to the list:
9:30 PM Sunday vespers at St. marks Episcopalian cathedral. It’s in an unfinished, dark huge cathedral from years ago – and lots of people from all walks of life come to see hear the chanting and all-mens choir; they bring blankets and camp out on the floors. It’s cool, historic, slightly eerie, short, and definitely unique.
@intparent , OMG!!! I didn’t realize that the forbidden T word was crossed out!! So funny! If you didn’t tell me I would never have known!!
@eandesmom We found public transit to be pretty good in Seattle. The light rail was a breeze to use and just recently started going up to Capitol Hill and to UW.
@LasMa if you are sticking to the downtown area or places along the light rail, it can be ok. However, OP was asking about getting out to one of the islands, or out of the city into something more rustic/cozy. There is that in spades in many many places here however getting to most of it without a car is difficult at best. While getting to a ferry itself would be very doable, any of the ferries really as the train does run along the waterfront all the way to Vancouver, getting to a cute inn or B&B on the other side without a car is more difficult. Most of those types of accommodations are not going to be near the ferry dock. If OP is willing to take a bike that opens up a ton more options but once on the other side, the public transportation options will be much more limited.
It isn’t impossible, Friday Harbor in San Juan Island would have by far the most options closest to the ferry dock but would take a chunk of time to get there. Kenmore Air (seaplane) could get OP there quickly from downtown but it isn’t inexpensive. However if money were no object, a great trip could be to take Kenmore air to Friday Harbor, stay there for 1-2 nights and ride the ferry from there around the San Juans. Or do a day trip from there to Canada. The main part of Friday Harbor is walkable from the airfield. and from the ferry dock. It is summer though so options may be slim. Still though, a bike will allow you to see more around any island itself, even if rented up there.
Same issue with some of the fun accommodations more in the mountains, such as Salish Lodge at Snoqualmie Falls which would be a very romantic get away and only 45 minutes from downtown. Hard to do with public transportation. There probably is a bus but it would take some research, uber would be pricey.
Woodinville with all it’s wineries has some nice lodging options as well. There are drive around services that could work if wine tasting was of interest.
A day trip on the ferry can be fun just by itself though, from downtown, Bainbridge is the prettier route in my opinion.
Seattle has a new entry in point-to -point car sharing company, ReachNow. Download the app, and find the closest car. Drive where you want to go. Leave and lock the car in any legal street spot. Like Cars2Go but nicer vehicles. My car-free kid used it yesterday and loved it.
You do have to keep the cars in Seattle, no trips to the mountains, but great for getting to spots off the bus/light-rail corridors.
There’s also a bike-sharing program. I’m not sure how that works or where you can go with the bike, but I did notice the green bike depots.
eandesmom, you’re correct that once you get out of Seattle proper, the transit becomes dicier. Although my D and her BF, who live in S.Sea, regularly visit his family in Snoqualmie which involves one bus change. I wish we had anything remotely like Seattle’s transit system here in Sacramento.
Honestly, if you’re going to spend the money for Uber or Cars to Go, just rent a car for a day or so. You get unlimited miles and you CAN leave town in them. If you’re not looking to leave Seattle at all, the bus, light rail, streetcars AND a water taxi and a monorail too can get you pretty much anywhere you want to go.
Driving and parking are a nightmare though.
An additional couple of restrictions are that my boyfriend can’t drive OR ride a bike (he can fly a plane for what that’s worth), which is a bother. I can drive, but I’m not a fan of driving and I’m a few years too young to rent a car from most places, anyways. 
That’s going to limit your ability to get out of town, then. But I know for sure you can find some nice places to stay on Bainbridge right off the ferry as H and I have done it without a car. But there are some boutique hotels or B & B’s inside Seattle, but they’re not cheap though! So be aware of that.
I would say then for a ferry destination Bainbridge with something walkable, or Kenmore Air (or other seaplane operators) to Friday Harbor will be your best bet. There will be more to see in Friday Harbor itself town wise I think that is walkable but I’ve not actually stayed in Bainbridge to compare. You could take the train to Anacortes and then the ferry from there to Friday Harbor but I’ve no idea how close the train station is to the ferry dock. Likely close but I am not sure. Bainbridge would be easiest, though more limiting.
There is so much to do in the city though that isn’t “touristy” per-se you really could just do a day trip. Ride the ferry to Bainbridge, walk around a little, have lunch, ferry back. The highlight is the ferry ride. @HiToWaMom has a great list of neighborhoods. I really would stay in the city if you only have 4 days, with a one day ferry ride built around it. You could end back at the waterfront and have a lovely dinner down there or something at SixSeven or another waterfront location for your birthday, all walkable.
In summer accommodations anywhere will not be cheap, the earlier you book the better.
@LasMa I can honestly say that is the first time in my life I’ve heard anyone say we had a good transit system! LOL! But it is all compared to what you are used to. Having lived elsewhere and seen what can work, ours is terrible. They are trying but there is a long ways to go. Some things have improved a lot and ultimately yes, you can get almost anywhere but for example from my house, I’d be surprised if I could get to Snoqualmie in less than 2 hours by bus (and it would be 2-3 busses)…but 45 by car.