From US12, the drive down FS25 to the east side of MSH will take up most of the day. 25 is paved, but not fast.
For Rainier, from 12 I'd approach it from the SE side, and drive up to Paradise - to see the most scenic part of the mountain.
For Olympic NP, you could take 101 in either direction. If you are going to spend most of your time on the north side, and exit via Port Townsend (and the ferry to Whidbey), I'd suggest 101 up the west side, getting a fuller taste of the rainforest side of the mountains and park.
The north side, especially around Sequim is on the rain shadow side of the Olympics. Our biggest winds and storms have SSW winds. So the west side will always be cooler and wetter.
In Seattle we say that summer doesn't start until July 4th. Then we have two months with little to no rain. Chance of rain increases in September, peaking in late November.
Heading back east, consider WA20 as an alternative to US2. It's a somewhat more scenic route over the Cascades (not that 2 is bad).
Google Maps and its streetview lets you preview most your route. WSDOT has a good collection of webcams. A UW prof, Cliff Mass has a good blog about area weather patterns. He has high praise for the newest weather radar on the SW side of the Olympics that lets them see storms coming in off the Pacific.