While many beginning kayakers are worried about mastering the roll and what to do if they can't roll and need to get out of the kayak, the paddling stroke is equally important. Efficient paddling will help keep your body relaxed and less fatigued, and help you avoid potential trouble spots on the river.
Most experienced paddlers prefer to use kayak paddles with offset blades. To set up a paddle for offset use, while looking at the paddle, push the button that lets you split the paddle in half and rotate the right blade so that it faces you if you are right handed, or the left blade faces you if you are left handed. Once you are in the boat, hold the paddle with your dominant wrist fixed and a loose hold with your non-dominant wrist, and keep your back straight.
Place the blade of the dominant hand in the water at a 90-degree angle to the kayak and pull the blade back toward the boat, then lift it out of the water. This motion pushes the other blade forward to the point where it can be dipped in the water. Don't stick the paddle shaft all the way into the water. Keep the stroke simple, just dipping the blade in. You will notice as you start pulling that your arms are making a figure-8 motion with each stroke.
Once you have the basic motion down, start to work on using your torso more in the stroke to conserve energy for long sessions on the river. A simple way to practice using your torso in the stroke is to lock your arms at the elbows so that you are forced to reach for each stroke with a twist of the upper body, instead of just with the wrists and shoulders.
There are times, especially where speed of paddling is important, that just using your arms and wrists is better, but for general paddling, use your torso as well.