Starting to plan a road trip
Hi, and welcome to RTA!
First thing you need to do, to plan a road trip, is to get hold of a road atlas or some detailed US maps. Since your trip isn't until April, you could order an atlas from the RTA store, if you can't get one locally. That would give you a good start.
There are 59 national parks in the USA, and most of them are in the West. But there are quite a few in the eastern area. There are 119 national monuments, and they are spread out across the US. All of them will be on a good national map in an atlas, and listed on the individual state maps in the atlas. So will some of the larger, or more meaningful, museums and theme parks as well.
Once you get that map/atlas, you can start to flag some of the spots that you'd like to see. Go with an east coast city that has reasonable airfares. If it's NYC, don't rent your vehicle until you are done seeing the city, as a car is a liability there (as well as in Boston and Washington DC). If you rent once you're done in the first city, you can arrange to stay outside of the other cities in a place with access to public transportation, and use that to get into the city.
Bear in mind that renting a vehicle in one city and dropping it off in another will likely cost you a hefty fee, called the One-Way Drop Fee. Sometimes, if you arrange your rental via an overseas consolidator, they can get those fees waived. Otherwise, with 3 months, you have an ideal amount of time to actually make a huge loop in the US and drop your vehicle back in the original city of rental.
Iconic places -- Grand Canyon, Yellowstone, Zion, Arches, Rocky Mountain, Yosemite, Everglades, are all well known national parks and considered by some as things you should see in your lifetime. Consult the National Park Service website which will contain links to each of these, and about 50+ others.
Other planning things: To make your trip more enjoyable, plan to drive 500 miles per day or less. Plan to spend at least one day in each national park (Yellowstone needs at least 3 days, as it's huge). Get reservations for hotels in/near the parks as soon as you get your dates, particularly for any of the above parks as they are the ones that reserve quickly. Same if you plan to spend time at any of the US theme parks, particularly DisneyWorld (FL), Disneyland (CA), Cedar Point (OH), Universal Studios (FL or CA) which is where Harry Potter Worlds are each located, or Six Flags Over Mid-America (St Louis, MO). This is especially true of the theme parks on the weekends any time of the year!
Donna