Building strength in your arms is not a necessity for becoming a great golfer. Learning the proper grip, swing, and stance are more important in the overall scheme of things. However, if you build strength in your arms and learn the other skills needed to become a good golfer you will get more distance and should become better overall.
Start out with dumbbells in each hand. You should have a weight that you can lift comfortably. Start out by holding the dumbbell perpendicular to the ceiling. Lift the weight to shoulder height, turning your palm inward so that the dumbbell is parallel to the ceiling. Return to the original position. Do the same thing with the weight in your left hand. Do 10 lifts with each arm and then repeat the set.
Do an overhead extension to build strength in your triceps. Take a dumbbell and hold it in both hands behind your neck. Your right hand should be on top of your left. Lift it straight up and then return it to the starting position. Do 10 lifts and then switch your hand position so your left hand is over your right. Take a 30-second break and repeat the set.
Perform tricep dips on parallel bars to gain arm and hand strength. Support your weight with your arms as you hold your body up on the parallel bars. Bend your elbows to a 45-degree angle and then push up to a straight arm position. Do this 10 times, take a 30-second break, and then repeat the set.
Do the bench press to build upper-body strength and gain more power in your arms and chest. This needs to be done with a spotter. Take a barbell and hold it at chest level. Press it up until you have locked out your arms. Return to the starting position. Do this 10 times and then take a 30-second break and repeat the set.