I will echo Jameel's thoughts that late July is perfect. I have gone this first week in July in the past and found Reflection Lake still snowed over, and snow on a lot of the trails in Paradise, so I learned the hard way that the last week in July into the first week in August is the best time for access and wildflowers.
Mt Rainier has lot's of places to get in, and go around. My favorite is the Paradise area. Hands down it's the best area of the park. So many awesome trails. And while summer time is a busy time there, I have never had an issue finding parking or hiking. There are other people on the trails, but it's not a big deal as there are so many trails.
If you can get a room in Paradise Inn. That hotel is like 100 years old, and so it has that old world charm to it. I have stayed there a couple of times and would love to get up there to stay again.
Olympic National Park, the tip for there is to look at the tide charts. They are best when the tides are in the lower cycle, and then of course when it's low tide so that you can access the beaches and get great reflections. When it's high tide a lot of the beaches can disappear. So as you put this together, and you are splitting days between multiple locations look for when the tide is low at sunset or sunrise if possible and pick those for your days in the Olympics.
