I'll give you a short answer for now - for most out of towners

late July and August would probably be your best bet for weather. You should try to time it for the alpine wildflowers in Rainier and Olympic NP. First 2 weeks of August is usually when wild flowers peak but it could vary year to year.

Don't go in winter. The mountains are all covered under snow, making driving and sight seeing difficult.

It would be much more fun to play in snow in shorts and Ts in summer for most tourists, which is almost always possible in Rainier.

If you have more questions, I will respond when I am back in town this weekend. Too busy and tired this week.