Welcome aboard the RoadTrip America Forums!
If I were going to make the trip that you've outlined, I would probably start with Yellowstone and finish in Seattle. Yellowstone will be busy throughout the timeframe of your planned time on the road but only gets busier through July until its visitation numbers peak in late July through August. Meanwhile, in the Pacific Northwest, an area known for its rain, average precipitation continues to drop until reaching its yearly minimum around mid-July. Neither of those factors is guaranteed or a deal breaker even if they both work out, still you should plan on making the best of what conditions are usual.
There are a couple of small airports near Yellowstone with at least seasonally scheduled service, including Billings and Bozeman in Montana, Cody and Jackson Hole in Wyoming, and Idaho Falls and West Yellowstone in Idaho. Prices to fly into such smaller airports tend to be higher but can fluctuate quite a bit, so it's worth a bit of your time to comparison shop. Don't forget to also check the cost of a one-way rental from each airport which can also vary considerably. The nearest large airport is Salt Lake City UT, but that's about a seven hour drive from Yellowstone, more than you can cover in a single day after flying.
After Yellowstone, I'd make my way west through the Snake River Valley on I-84 to Portland, then up the Pacific Coast (US-101) and around the Olympic Peninsula (Olympic National Park t Port Angeles, cross over to Victoria for my time there, catch another ferry to Vancouver and finish up by making the drive from there down to Seattle. That minimizes your driving time, but driving will still eat up the equivalent of a minimum of three to four full days out of your vacation time depending on ferry-boarding and border-crossing wait times.
Places I've enjoyed and can recommend more or less along the route outlined above include Bruneau Sand Dunes and Bruneau Canyon in Idaho; the Historic Columbia River Highway (old US-30), the flower gardens of Portland, and the reconstructed Fort Clatsop in Astoria OR; and the harbor area of Victoria and Butchart Gardens north of town.
AZBuck