There are a number of companies that rent RVs in the USA. Cruise America is one of the larger ones. You can find a selection of companies if you just do a search on "RV rentals". I suggest that you pay attention to the locations of their offices. Some of the smaller rental companies may only service certain regions of the USA.
The cost of an RV rental can start at $100 to $150 per day for a smaller one and go up from there. Remember to include the cost of mileage which can become quite significant when touring the wide open spaces of the West. In addition to the rental price, you must pay for the fuel, which can be quite costly for some of the gas guzzling vehicles. You must also pay for the campground fees that could amount to $30 to $50 per night with full hookup. Your total cost for transportation and lodging will be around $150 to $200 per day. If you compare this to a budget car rental with its better fuel consumption plus cheap hotel rooms, the RV may not be the cheaper alternative.
There are some ways to save money while camping with an RV. You can stay at the campgrounds in the National Parks and State Parks where the fees are often much cheaper. You might have to forgo the hookups for electricity and water, but you can typically find a dump station to take care of your waste water. You might also be forced to forgo the hot showers and swimming pools offered by the commercial camp grounds. You can even find some places to park the RV overnight for free. Most of the Walmart stores across the USA allow free overnight parking for RVs in their parking areas. Some budget minded RV campers will spend several nights at free areas or at inexpensive public campgrounds, but spend one or two nights a week at a commercial campground where they can find hot showers, swimming pools, laundry facilities and waste water dumping stations.
Aside from the travel and lodging, you will not have many major expenses. Food is relatively inexpensive in the USA. If you cook most of the meals in your RV, you can easily budget $50 or less per day for groceries for two people. Entertainment and attraction fees can be the biggest other expense, but most of that is avoidable if you maintain a Spartan lifestyle.
I don't think there are any particular safety issues. Be careful driving the RV on mountain roads, backroads, and crime-prone areas of the big cities. Campgrounds tend to be great places to meet friendly, helpful and gracious people.
The best time of year to visit most locations is in the summer months, but that is also the high tourist season when the campgrounds and attractions are the most crowded. The weather in the Autumn especially during September and October can be quite nice. In the northern states, it can be warm and sunny during the daytime and cool at night. In the Souther states and in the deserts, it is still very warm but the oppressive heat is a bit moderated. Best of all, the campgrounds and attractions are much less crowded at that time.