1.) Try not go faster than 5mph over the limit. NPS law enforcement does not play around and they know where to sit. But like PandaBear said, don't expect to drive very fast either. Too many people slow down and cause a traffic jam behind them just to take a picture. Use a pull-out shoulder if you want to stop and take pictures.
2.) To beat the crowds wake up early and get to your destination early. Yellowstone is one of the most visited National Parks in the US. The Grand Tetons aren't as popular, but the smaller size, 2 lane roads, and proximity to Yellowstone means it can feel just as busy.
3.) Bring bug spray, the mosquitoes and other biting insects are big and numerous enough to carry you away. I spray my hiking clothes and gear in
Sawyer Permethrin Insect Spray before I leave for a trip. They also have one where you soak your clothes so the Permethrin lasts longer.
4.) Stay away from wildlife and do not feed them either. Ignoring both will get you a hefty fine.
5.) Know what to do if you come across different types of bears.
6.) Be a traveler, not a tourist. Tourists suck; they're loud and have no respect for anything.
7.) Take nothing but pictures, leave nothing but foot prints.
8.) Check out the Beartooth Mountains while you're out there. A lot less people but a lot more camping options.