Top car-rental companies in the states are: Hertz, Avis, Alamo, Enterprise, and Dollar Rent-A-Car companies. All are cross-state hire companies. Be aware, however, that they charge a major "drop-off" fee (as high as $200-$500 extra) if you rent the car in one location, but drop it off at another location when you are done--so either do a "circle" route--or else figure out what the drop-off fee will be (and try to negotiate a rate with no drop-off fee).
As far as stuff to see and do, I suggest you start with a plan--want to see mostly cities? mostly national parks? all of the US, or just certain states and major cities? Looking to see the unusual? tourist attractions? primarily sites related to different films? want to get "back to nature"? or spend time in small cities to meet the locals? do you want to "hang out" in clubs and music concert sites? or at the beach? or in the mountains?--or some of all of the above?
There are lots of guidebooks--and we can help you with cities on this site--but we can't cover the whole US in detail in one post. So give us a general idea of interests--and then, over time, we can cover some of the details.
I'll give you details on Seattle for starters.
The Seattle airport is south of town. Near here is the Boeing headquarters where they build the planes, and also a plane museum which has planes you can walk through, including ones they didn't build (like a supersonic transport). If you call them early, you can even visit the factory while they are building planes.
In downtown is Pikes Place Fish Market (featured in lots of movies once such as Rock Star and The Fabulous Baker Boys). Also here is the Space Needle (featured in lots of movies, including the one called 10.5, which is about an earthquake that knocks it down). There is also a movie called "The Blob" where the Blob took over the Space Needle. Right next to this is the "Experience Music Project" which has lots of clips from music videos and movies, and also allows you to try and make your own music (using instruments they provide) and see original manuscripts of when famous music was written, and included in the same building is an "Alien Museum". A very short ways from here is Lake Union--where there are lots of great restaurants right on the lake. Just north of here is the University of Washington and a fun area next to it. Just south of here (east of downtown) is the Japanese tea gardens and the Arboreteum--definitely worth a visit. Heading east from here you will cross Lake Washington (10 miles across) on the interstate. The city of Kirkland is on the eastern side of the lake, and is a fun place with restaurants and clubs. Just northeast of here is Redmond where Microsoft's headquarters is located and where there is a great shopping mall and next to this is a fun group of shops to visit. Heading out east from here is the giant waterfall at Snoqualmie (higher than Niagara Falls), and a nice area around another large lake. Just north of Redmond are some good wineries near Woodinville, including Chateau St. Michele.
West of downtown is the large Puget Sound--and if you go south and then west you can go to Alki Beach (featured in the movie "Sleepless in Seattle"). North of Seattle (on your way to Vancouver, Canada), you can go west to Anacortes and take your car on the ferry across to Vancouver Island and the city of Victoria, travelling through the beautiful San Juan Islands on the way. In Victoria, visit the harbor and the old British town next to it--and then travel north to see Butchart Gardens and a place called Butterfly World. Then head north and take your car on the ferry to Vancouver. In Vancouver, viist the area around the inner harbor, and also the ski resort called Grouse Mountain (just as good in the summer as in the winter). If it is winter, go north to the ski resort of Whistler (about 30 miles north). In summer consider going east through Banff to Calgary. Also, you can go south back to the US and down to Portland. Lots of movies and TV series were filmed in Vancouver--at one time it was estimated that 20 US TV shows were being filmed at the same time there (back in 1999)--but that was when the Canadian dollar was real weak. Now it is strong, so more are filmed in Hollywood or New York now. Some that were filmed in Vancouver are X-Files, Poltergeist, Police Academy, The Net, Stargate SGI, The Addams Family, The Sentinel and McGyver.
P.S. While in Seattle, it rains a lot, and when it doesn't, you will be able to see the snow-capped Mount Rainier--in fact the local saying is "If you can't see Mount Rainier, it is because it is raining, and if you can see Mount Rainier, then it means that it is about to rain".
Enjoy