There is no one "correct" procedure for connecting to a RV campground's utility hookups. The important thing is to develop a routine and an awareness that prevents contamination, avoids electrical shocks and keeps your feet dry.
Before going through the trouble of backing into a campsite and leveling your rig check the site's electric, water and sewer connections. Check the electric outlet's polarity and voltage, the water's clarity and aroma, and the sewer inlet's opening to be sure it is clear of rocks, cans or debris.
While you are there, rinse off the RV campground water hydrant and spray it with a household disinfectant. You never know what the previous occupant or his dog has left on that hydrant. By the time you have finished backing and leveling your rig, the disinfectant should accomplish its purpose.
Once your rig is in position and leveled, you can go through your hookup routine. First, plug in the electric cord while your hands and the ground are dry. Turn off the electric outlet's breaker switch before inserting or removing the RV's electric plug. This will prevent shocks and avoid damage to the prongs of the plug.
Next, attach a water-pressure regulator to the campground hydrant so it can protect both the drinking-water hose and the RV's plumbing from sudden surges in campground water pressure. A water filter (to remove sediment) is next, followed by the drinking-water hose. Run water through the hose to remove the air then attach the hose to the RV.
By the way, all of the water-hose connections can be made quick and easy with the use of "quick-connect" fittings on both ends of the water-pressure regulator, water filter and water hose. These handy fittings can be found at RV accessory stores like Camping World and in hardware stores where water hoses are displayed.
The sewer hookup is last. Once the hose is connected, double check to be sure the sewer hose is properly secured to both the RV and the sewer inlet. Think seriously about wearing disposable polyethylene gloves when handling the sewer hose. They come in boxes of 100, are inexpensive and can be thrown away after use. Some RVers prefer to use disposable, latex surgical gloves. Even if you wear gloves, it's a good idea to wash your hands with antibacterial soap immediately after handling the sewer hose.
The gray water valve may be opened after the sewer hose is connected. The black water valve, however, should remain closed until you are ready to dump. This assures a buildup of liquid in the black water tank. The more liquid in the tank the better the flushing action will be when the valve is opened. A day or so before dumping the black water, close the gray water valve. The captured gray water can be used to rinse the sewer hose after you dump the black water tank.
Dumping the holding tanks is pretty straight forward. Black water is dumped first, the valve closed and then the gray water is dumped. The gray water does a good job of rinsing the black water out of the sewer hose. Lifting the RV end of the hose drains the remaining water into the campground's sewer inlet.
After closing both the gray and black water valves, the RV's sewer outlet is capped and the sewer hose stored in the sewer hose compartment. Run the kitchen faucet (using the rig's water pump) to add a couple gallons of water to the gray water tank. Use a two-gallon bucket to pour a few gallons of water through the toilet bowl and into the black water tank. This will prevent any solids from turning into a low-grade cement in the bottom of the tanks. This is also the time to pour in any holding tank additives.
Disconnect in the same order you hooked up; electric first, water next and sewer last.
The drinking-water hose and fixtures should be stored in a compartment separate from the sewer hose and its attachments. Both ends of the drinking-water hose can be connected together to keep moisture in and critters out.