Glad to hear you're considering a road trip across the USA. If you decide to go through with it, you will not be disappointed.
You definitely want to rent a car. It is by far the best, most efficient way to see the country. It offers you much more freedom to come and go as you please than having to rely on public transportation. In America, the car is THE way to travel! It takes about 5 days to travel from east to west, but that's with minimal stops.
There is so much to see and do. The country is vast and has something unique to offer each individual's personal interests. How much time and money will you have? If those won't be issues for you, then I'd suggest taking your time and really enjoying the trip as much as possible.
I'm a truck driver, so I think I can speak somewhat confidently on this. I'll try my best with the limited information I have on you, and you can pick and choose what appeals/works for you.
If you just want the experience of driving from the east coast to the west coast, you can drive Interstate 90 in the northern region from Massachusetts (MA) to Seattle, Washington (WA). This is my personal favorite because of all the natural beauty you'll see along the way.
Or, you can take Interstate 70 in the middle region which starts in Washington, DC and leads to California (CA). This is a great way to go as well--it offers beauty, but it's different from the North. You'll see a lot of farmland going this way.
Or, you can drive Interstate 10 in the southern region from Miami, Florida (FL) to LA. If you like the heat, deserts, and the gulf coast, this would be the way to go.
All 3 ways offer a beautiful and unique trip--I guess it just depends on what you like.
Driving Highway 101 from San Diego, CA to Seattle, WA all along the west coast is a treat in and of itself. There is stunning natural beauty and beaches all along the way. You obviously only get to see the west coast, but if you're limited on time and money, you can't lose with this choice.
The must-see cities are New York, Los Angeles, San Francisco, Chicago, Seattle, Savannah, Austin, Dallas, Las Vegas, and Washington, DC. Washington is one of the most beautiful cities I've seen with all the monuments and museums. The traffic around the city is horrendous, though.
From my experiences, the states with the most natural beauty, in no particular order, are: California, Oregon, Washington, Maryland, Wisconsin, Arizona, Montana, New York, Maine, Vermont, Pennsylvania, Florida, West Virginia, Virginia, Tennessee, Georgia, and Colorado. However, each and every state has its own pockets of beauty. You just can't always see everything from an interstate.
As themachine stated, the natural wonders are an absolute must-see. He gave you a link to help you there. The Grand Canyon in Arizona is breathtaking.
Yes, America is a big country, but if you have the time, the hours spent driving through deserts/prairies/farmland to your next destination are where a lot of the fun is. Just take in the sights because you never know if you'll ever be back. And let's not forget the people--I hope you'll meet lots of wonderful folks along the way.
Remember, as in life, it's the journey that's special, not always the destination.
I hope this helps--feel free to message me if you have any other questions.