Google maps shows I82 - I90 - WA18 - I5 - WA16 (bridge) as shortest, by about 50 miles. Choosing a departure time of 9am weekday it shows potential traffic issues on the I5 piece through Tacoma.
This route also has several passes - I82 north of Yakima, I90 over the Cascades, and WA18.
I205 and I5 past Portland is longer, and can have traffic issues (though it largely bypasses the cities), and doesn't have any passes. I5 Olympia to Tacoma can be slow, but that's mostly commuter rush hour traffic in/out of Tacoma.
Avoiding Tacoma and the Narrows bridge by way of Shelton and WA3 adds 20 miles and half an hour to that southern route. It replaces the city freeway with 2 lane semi-rural roads. WA302 is curvier.
GMaps has streetview for the whole route. WSDOT website has traffic cams, alerts and construction notes. So you don't have to take our word for any of this.
Tacoma Narrows tolls are collected east bound (away from Gig)
Tacoma area construction news