For home rentals I'd look at the nearest residential areas that surround downtown. Places like Queen Anne, Fremont, Ballard, Wallingford, Ravenna, Capitol Hill. Each of the neighborhoods has their own "flavor" and plenty of shops, restaurants, access to public transportation, exercise opportunities, etc.

Besides assuming you'd also be spending at least sometime in the downtown area doing the tourist things, with plenty more restaurants, shops, activites.

A car for a day trip to Snoqualmie Falls &/or overnight to Whidbey Island are beautiful but easy suggestions outside Seattle.

Residence Inn Lake Union.

Nice location, up and coming neighborhood, Light rail outside door into downtown, with Whole Foods less than 2 blks away (can get prepared foods, nice seafood etc) for groceries (RI has a kitchen.) Easy freeway I-5 access.

Book a lakeside view patio and watch the view of the Lake and float planes.

LAke Union has a nice around the Lake walking/jogging path, Dukes, Chandlers restaurants right across the street and outside patios on the LAae or do take out. Capital Hill with all of it's restaurant variety nearby.

Near to Seattle waterfront, Pike PLace etc.

If you stay on Capitol Hill, you will have many dining, shopping, and entertainment options and you will be within a 1- to 1 1/2-mile walk of downtown and all that it has to offer.

HTTY

At the end of your time in Seattle, take an Amtrak train to Portland. Rent a car in Portland or PDX and go out to see the waterfalls in the Columbia Gorge. Fly home from PDX.
The TriMet Light Rail runs from downtown Portland out to the airport.

You can also side trip to Vancouver... its a few hours away

There are plenty of things to see/do in the Seattle area. If you're flying to Seattle you will want to rent a car so you can see the sights easily.
Here are some ideas:
In Downtown Seattle: Pike Place Market; Underground Tour; Duck Tour; Seattle Center/Space Needle/Experience Music Project; Safeco Field for a Mariner's Baseball Game.
Future of Flight (Boeing) Tour in Everett, about 30 miles north of Seattle. Couple this with a ferry ride from Edmonds to Kingston for a scenic little trip.
Mt. Rainier National Park
Northwest Trek (near Mt. Rainier)
Mt. Saint Helens Visitor Center
Museum of Flight, Boeing Field just south of Seattle
The Arboretum & Japanese Garden
Takke the Victoria Clipper to Victoria, BC or the San Juans for a few days.
Olympic National Forest on the Kitsap Peninsula.

We also stayed at Residence Inn, Lake Union - but it was about 10 years ago, so I was hesitant to recommend it. It was perfect at the time. I am not sure you could fill 2 weeks of afternoons with things to do, since some of the day trips mentioned above would take a full day.

I would consider perhaps 8-10 days in Seattle and the rest of the time in Vancouver, BC. Almost more to do there - you will not need a car since a great walking city and great public transportation (assuming you either have or want to get a passport)

Going up to Victoria, on the Victoria Clipper, passenger boat leaving right from downtown Seattle, makes for a nice (long) day trip or 1-overnight.

Another vote for the beautiful and convenient Capitol Hill neighborhood. There are some REALLY nice VRBO listings for that area. Volunteer Park is a few short blocks north and has gorgeous view from the tower.

Also, South Lake Union (where Residence Inn is located) is not much of a residential neighborhood and, though it is not and overly long walk from there to Capitol Hill amenities, the walk is up a steep incline on some busy or otherwise unpleasant thoroughfares.

HTTY

I would definitely stay 1-2 nights in Victoria. There's so much to do/see. The trip up there takes about 2 hrs. Butchart Gardens is a "Must-do" in Victoria.