Tennis elbow is common in adults who have never played tennis before. People over 40 years old are most likely to get tennis elbow. It could be you are using a racket that's too stiff. Have someone with good experience check this out for you.
Studies show players who hit the ball late or who "frame" the ball are more likely to get hurt. Framing the ball means the player has miscalculated the swing and hit the ball with the frame of the tennis racquet instead of the strings. Ask someone who plays tennis to watch your swing and offer suggestions.
Every physical activity or sport requires strength in certain muscles. In tennis, weakness in the muscles of the forearm can lead to tennis elbow or other injury. Make sure you are warming up properly before playing. An exercise program to build up strength is also a good idea.
You may want to consider taking some lessons through your local parks and rec department, a tennis club, the YMCA, or other health club in your area.