It costs to get into most national parks. Calculate if it is cheaper to buy the annual pass (be sure to get the one that covers ALL parks) or pay each time. In general you are better off with the annual pass.

Loop the trip so that you visit the warmer (southern) parks first. Your trip timing is at the cusp of summer. A few weeks can make a big difference.

If you get to the parks in the morning (instead of end of day) there are more available campsites, and you can also get an overnight backpack permit and just start your backpack, so avoid camp fees at the campgrounds. If you want to do some of the more popular trails, reserved permits are needed. However, if willing to take some less used trails, walk-in permits are usually available for one person. I rarely reserve permits.

Do your homework. If you have the access and details nailed down you will waste less time just going around in circles being confused.

Realize that speed limits within parks are really slow. Yellowstone in partiular, is large, and do not expect to make very good time on the roads. An average of 30 mph is about what you can expect, without traffic.

When I do a combined car-camping/backpack trip, I usually have a minimalist backpack ready to go with its own set of clothing and the little stuff (personal gear, etc) already packed. I have a separate ditty bag of personal gear for car-camping. That means doubling up, but it also makes it fast and easy to head out for a backpack without sorting through all your stuff. For overnight backpacks, I would not bother with cooking - just take non-cook stuff. This will simplify your efforts.

In parks where food storage is required be careful with not having too much food in the car and keep it in easy-to-carry containers. You have to move all that food into storage boxes that are provided in the campgrounds. Smelly hygiene items (toothpast, etc) also have to go into animal-proof storage.

I would pack pretty minimal for the car camping too. After a few days you can see what you would like to add and simply buy that stuff along the way.

Truck stops are great places to take a shower. Also, many gyms will let you shower for $5. And public swimming pools (unfortunately they may not be open yet), just pay for swimming (usually a few dollars), skip the swim and shower.