The answer depends very much on your personal preferences and interests.

If you love culture: New York City is at the top of the list, with all its museums, Broadway, Radio City Hall, etc. You can also add other major cities such as Boston, Washington DC, Chicago, San Francisco, and Los Angeles.

If you are a nature-lover: Explore the major National Parks such as the Grand Canyon, Yosemite, Yellowstone, Zion, Grand Tetons, Niagara Falls, etc.

If you love night-life and action: Make sure Las Vegas is on your list, along with New Orleans, and probably NYC and Los Angeles again.

If you are into politics: Washington DC is your best and almost only bet.

If skiing is your thing, the best skiing is in the Rocky Mountains - e.g. Aspen, Vale, etc.

If you love amusement parks and theme parks, the most likely starting point would be the Disneyworld area, with Universal Studios next door, and Cape Canaveral nearby as a bonus, though there are many many more across the US.

Etc.

In general, if you're more into things having to do with human presence and activities, the East Coast, and especially the strip from Boston in the north, moving south through NYC, Philadelphia, Washington DC, Baltimore, and all the way down to Atlanta.

If your preference is the wide open and mostly untouched spaces, you'd be more interested in the area around the Rocky Mountains plus a bit to the west and a lot to the east.

Most people will probably say New York.

Personally, I'd go with Los Angeles.

In LA you have:

Awesome beaches
Amazing restaurants
Scenery ranging from beach to mountain to desert
Attractions like Disneyland, Magic Mountain, Knott's Berry Farm, Raging Waters, Universal Studios
Fun places to take the family like Family Fun Center and Medieval Times
Chinatown

Not too far from Mexico and you can get to San Francisco in a few hours. Sick of the city? Head to San Bernadino to get lost in a pine tree forest or visit Santa's Village.

Nothing compares. The Grand Canyon? It's a big hole..., Rushmore? Have you ever actually been to the Dakotas? I'd rather spend the night in jail haha.

LA is the best way to spend 5 days in the US. Grand Canyon is nice for one day or if you have social anxiety disorder and can't stand crowds.

What I recommend is this:
Day 1: Drive to San Diego, visit Balboa Park (San Diego Zoo optional). Take the trolley to Tijuana and eat as much food as you can (it's delicious), buy some cheap silver jewelry and take in the local sites.
Day 2: Visit the beaches until you feel you're getting too sunburned. Go to Santa Anita racetrack and bet on some horses. Finish in Chinatown for dinner.
Day 3: Hollywood. Take a tour, visit the local shops.
Day 4: Disneyland or Universal Studios
Day 5: Drive to San Francisco and visit Alcatraz or something. Your legs will be killing you after Day 4.

It really is personal preference. I can't stand boredom so I thrive in places like LA and Orlando. New York seems overcrowded and doesn't offer enough of what I like as silly as that sounds.

I would say stick to the east coast. I personally would stay in NYC, hit the touristy Liberty Island, eat at the stage deli, catch a broadway show. Spend a couple days there.

Then drive over to Boston, and spend a couple days there. Eat some chowder at the Union Oyster House, walk on the Freedom Trail, Fanueil Hall, Fenway Park if you're a baseball fan.

Or if you like politics or musueums, landmarks drive down to DC and spend a couple days there. Walk around the capital, Washington Monument, drive by the White House, go to the musuems.

There's just a lot more history and landmarks and culture in the Northeast.