I would go to Puyallup on Wednesday, spent the night or if you got into Sea-Tac early drive to Eatonville and spend the night. There's a lot to do and see up there.
We go to Mt. Rainier a lot. We live 45 miles away so it's pretty handy. We like to go up for a picnic dinner a couple of times a month at one of the campgrounds in the summer. Believe it or not, I used to work with people who had never been to Mt. Rainier NP, they were born and raised in the Puget Sound area.