In the US, this sort of vehicle is known as an RV, or recreational vehicle. You are allowed to drive them on a category B licence from the EU, though you may need to be 21 years old. (You will need a BE licence if you want to drive a car or light pickup truck with a trailer attached.) You will also need your International Driving Permit.
It's not usually allowed to park on public streets and highways overnight, though this varies widely by city, and sometimes there are places within a city where RV parking is allowed or prohibited. This is so variable that it's almost impossible to provide a list.
Generally in the US, however, you can always park on private property with the permission of the property owner.
Walmart stores in the US (a chain of hypermarkets not dissimilar to Hipercor or Carrefour) are widely known for permitting RVs to park overnight, though not every store allows this, based on local conditions. Check with the store manager to ensure that it is OK to park overnight at each store, and if there is a specific location where you should park. Almost always this will be at the far end of the parking lot from the store.
Also, most Interstate highways (autopistas) have roadside rest areas where you may be able to park. These are usually 50 to 100 km apart, but they can be farther apart on occasion. (For instance they are almost never present in urban areas.) Sometimes they have waste water dump facilities for RVs. I understand that these are rare or nonexistent in your country, so you may want to visit one for the sheer novelty. Watch the signs as you enter a rest area; sometimes RVs park with the lorries, and sometimes with the cars.
But don't park in a "Scenic Overlook" any longer than it takes you to enjoy the scenery.
Most truckstops in the US also have dedicated RV parking and sometimes waste water dumps. They also offer laundry and shower facilities, for when you really need to get the road dust off you and your clothes. You almost always book these services at the truck drivers' fuel desk, though you will usually fuel your RV along with the cars and pay where they do. Some truckstops have dedicated fuel islands for RVs. You can almost always park overnight at a truckstop if you are refueling, and sometimes even if you aren't, but truckstops can be noisy at all hours, making for a difficult sleep. Again, watch the signs as you enter; while RVs sometimes enter at the same entrance as lorries, and sometimes at the same entrance as cars, they almost always park in a different area than either.
As you noted, you can also park at campgrounds which have RV sites, though virtually all of these will cost you money. Campgrounds in national parks and forests, and facilities provided by individual states, are almost always cheaper than private campgrounds, but tend to be booked well in advance. For national parks in particular, you also have to pay a separate park admission fee to most parks, so in such a case a private campground may yet be cheaper, unless you wanted to visit that national park anyway.
One unusual such private campground is the Circus Circus RV Park, part of the Circus Circus Casino/Hotel on the Las Vegas Strip. If you're heading to Vegas this will be very convenient.