It doesn't take a special bike to ride but low gearing is a must-have. Touring style bicycles with a triple and a low range of at least 28 - 28 or so are necessary in the high mountains. Even on pretty easy rides in California you might need a low gear because of very steep short sections.
Riding back into the San Francisco bay area I hit a one block long section of road that was maybe 18% in Vallejo. In San Francisco there are very steep hills but they are not very long.
Expect to ride normal touring speeds of 20 - 25 kph on the average if you are touring.
While crime is much higher in the United States than in Japan you can avoid most of the problems simply by staying out of the largest cities or areas with a lot of tourists who attract criminals.
Most people in the USA like Japanese though sometimes they're impatient with people who have problems with the language - nothing like in France though.
The Rocky Mountains are actually easier to ride in than are the Sierra Nevadas in California even though they are much higher. Altitude can become a problem though and there are passes in Colorado that are well above 3,000 meters. Oxygen can get rather scarce above that altitude and you can suffer from altitude illnesses as well. You should always plan any trips so that you spend your overnight stays below 2,100 meters to prevent some rather nasty possibilities.
Colorado is the ultimate Rocky Mountain riding. There are several places you might want to ride. If you're a mountain biker you can ride around Gunnison or ride Utah's White Rim Trail.
You hear a great deal about the Slick Rock Trail in Utah but it really is only a 12 mile loop that is extremely memorable mostly for the great heat and spectacular climbs and drops aided by the highest traction surfaces you can imagine. I don't like Moab, the town adjacent to the trail, because when I parked in one of the local restaurant parking lots the local teenagers came through in plain view going through people's cars!
As for touring - it is pretty hard to go wrong just about anywhere. Oregon is a nice place to tour. And the western side of Washington State is pretty nice as well though you have to plan your trips a little because of the roads being so crowded in western Washington.
But you could always ride the dirt/gravel John Wayne Trail from North Bend (about 50 km east of Seattle) to Vantage some 30 km east of Ellensberg, WA and pick up normal roads there eventually going to Idaho.
Couer d'Alene, Idaho may be one of the most beautiful spots on earth. Or if you ride out towards Lewiston and Clarkston you'll see an amazing natural wonder - the Snake River Canyon. And if you are not astounded by that you are pretty hard to astound. Lewiston, Idaho, is a seaport!
The Pacific Coast highway that runs from Washington to San Diego, California offers all of the hill riding you'd EVER want and the inland routes are a whole lot less advertised but between us I much prefer.
I rode through central California past rice fields. In one area there must have been a billion mosquitos per square foot. A couple of kilometers later there were a billion dragon flies eating mosquitos. Another couple of kilometers and there were millions of Swifts eating, no doubt, dragon flies. The ride along the river road on the Sacramento River was idyllic and quiet. There were parks along the way that allowed camping and had good facilities for the most part.
Drivers were for the most part polite and reasonably careful. Though you have to be extremely careful in harvest seasons because truckers make all of their money then and they drive like crazy to haul more loads and make enough money for the year.
I could write an entire book about travel in the western United States but you really can find a great trip just about anywhere just by throwing a dart at a map.