There are official ATV safety courses available with certificates of completion. However, they do not properly cover ride training in mountainous terrain. In fact, some teach you incorrect methods and safety.
In my opinion ATV riding is one of the most dangerous activities you can engage in. In fact, it is officially the 4th most dangerous sport in the United States. For this reason, it is imperative that you how to do it properly and safely.
Watch this video for a "brief" tutorial
This video goes over the basics. However, there is a lot of detail in there. Watch and listen multiple times. Every word of it is important. There are little details you should pay close attention too. For example: As you watch her drive the ATV down the rocky section, you will notice her body moving as she lets the ATV work under her. I can't tell you how important this is. Also, you might notice her left rear tire comes up off the ground just a little bit in one section. The importance of standing is huge here. If she were sitting, she could feel the sensation of the rear of the ATV lifting up. Some people don't react very well in situations like that when they are sitting. They may stop where the shouldn't, or lose focus on other important factors. Plus, like I said, if you are sitting in rocky or bumpy terrain, you could be thrown from the machine. Stand and let your knees soak up the bumps. Don't lock your knees either. You may as well sit if you lock your knees. Stay in an athletic stance with your knees and elbows bent so the machine can work under you without jolting you.
Speeding or riding crazy is the number one reason for accidents. I have literally witnessed hundreds of ATV accidents. I'm not going to tell you I know everything, but I do know accidents and what causes them.
Although the speed limit on National Forest trails is 20 MPH, speed is a personal choice. You decide if you're doing 1 MPH, 20 MPH or anywhere in between. This is the danger of ATV riding. I can't tell you what your safe speed should be in every section of a given trail system. Terrain changes every inch. I can tell you, your speed and level of cautiousness should be adjusted constantly.
All anyone can teach you with certainty are these factors:
1. Monitor your speed at all times. Adjust it when needed.
2. Ride cautious at all times
3. Stand up and keep your elbows and knees bent in bumpy / rocky terrain
4. Never put your foot down
5. Do NOT jump, do donuts, or drift around corners
6. Do not be under the influence of drugs and/or alcohol when operating an ATV
7. Always wear a DOT approved helmet, eye protection, and proper attire
8. Ride at a speed and on terrain you are comfortable with.
9. Lean forward, backward, and side to side as required, depending on the terrain and slope
10. Check over the proper function and condition of your ATV before your ride
11. Let someone know where you will be riding and be detailed.
12. If riding on public trails, always stay on marked (official) trails.
13. Pack out what you took in. Help keep our public trails clean and beautiful.