If you come in the summer, be aware that there are lots of people in the Valley. But you still get to see the great granite walls and waterfalls.
If you don't get permits in advance, just plan on standing in line to get the walk-ins and no-shows. Not sure of the procedures, but I am sure you can get that by calling the park service and asking. I think some are available the day before you hike, so you could get the permits, then hang around and see the sights in the valley, ...soak up the tourist experience.
If you have a wilderness permit, you can spend the night in the backpackers campground.
I'd still leave my car at Mammoth and take the shuttle into Yosemite Valley. Less parking hassle, and easier to reach when you complete your hike.
We'd love to read about your progress and planning as you work on this trip.