Tim, you should be safe just about anywhere. There are certain areas of large cities you might want to avoid unless you are in a tank. I have cycled all over the USA, well, not ALL over, but quite a bit. I have not encountered the first crime such as theft, robbery, assault, battery, and that sort of thing. All that and worse is going on, yes, but the touring cyclist is not likely at all to be bothered by it. I would advise using cycling maps. I assure you it can make a difference which roads you use. Some roads are as close to ideal for cycling as you can get. Others, especially at rush hour, can be roads from hell with noise, no shoulders, pollution, and mad-dash traffic.
Do not expect the same network of hostels as in the U.K. They are here and comparatively few and far between. The west coast will give you a good choice of cheap hiker-biker campsites. ACA's northern tier route is mapped coast to coast. Get the maps. Pay attention
to details. I would not tell anyone what touring bike to get. I am still using my old one and it works just fine. This past winter I cycled from coastal Florida to San Diego, CA in 55 days total with 43 days actually on the road. The frame may be old but it's efficient enough, and I am 60. I would recommend using Continental Travel Contact tires. Rotate when you are half finished with the trip, and one set should take you all the way across. Don't skimp on carrying patches and glue and tire levers. Don't go anywhere without a pump.