"Which areas" depend entirely on what you want to look at out your windows -- trees, bears, deer, office buildings, water, mountains, houses, farms??
For rentals in and around Seattle, this site called NWSource combines the Classifieds of the two leading newspapers in and around Seattle:
http://rentals.nwsource.com/rentals/
When you open that site, you'll see a map and a list of locations you can check. You'll be able to get a good idea of prices and amount of space available.
You might also want to check on Craigslist.
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Moderately-priced apartments
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