This horse is cantering - the legs move in a three-beat pattern, from back to front.
It's a lot smoother than trot - you end up kind of tracing an oval shape with your hips - but it can be harder to stay on at first as it's faster and requires you to be more secure in the saddle and relax. The more you relax, the less you bounce, and as a beginner I find that I have to concentrate to make sure my legs are gripping the horse tightly enough. When I started out, my instructors had me hold onto the pommel of the saddle with my hands and lean waaaaay back until I felt ready to let go, first with the inside hand and then with both.
It's lots of fun, if hard work while you're still breaking all your abdominal muscles in. Expect to ache quite a bit after the first few times!
As others have mentioned, there's something called the 'lead leg'. This is the front leg that moves as the third beat of the gait. In the image above, it's the leg closest to the camera. It's easier for a horse to move round a bend if they're on the 'inside lead' - that is, the foreleg on the inside of the curve moves on the third beat, stretching out on its own. There are specific commands to ask for a particular lead leg, but for beginners the easiest way to get the correct lead is to ask for the canter in a corner or on a circle, as the horse usually picks up the correct one all on their own then.
As a beginner, I ask for the canter by sitting the trot and squeezing or kicking. I'll progress to more sophisticated aids once I get the hang of that.