Prepare for movement with movement! The purpose of a warm up is to prepare the body for what you will do next. If you plan to partake in cardiovascular exercise, then you will want to warm up with a dynamic movement of sorts that is similar to the exercise you plan to do. For example, if you are planning to do some jogging, then warm up for a few minutes with a walk before transitioning to a jog. This is a good way to let the body adjust and respond to what you are about to do. If you then want increase intensity to perform intervals, your body will be ready. Avoid coming in off the street, hopping on a treadmill, and cranking up the speed to full blast!
If you are planning to strength train, then take your body through the same or similar motions with light resistance before hand. This is an excellent way to warm up the joints that will be moving while practicing good form and technique before applying a heavier load. For example, if you plan to do a dumbbell press, start with either no weight or light weights and move your upper body dynamically through the pressing motion while keeping your core muscles engaged. You can also do other movements like arm-swings forward/backward and laterally, and other shoulder warm-ups like internal/ external rotation and rolling the shoulders forward/ backward. Do 8-10 repetitions of each of these movements starting with a smaller range of motion then progressing to more as the muscles and joints become warm. Be sure not to bounce forcefully or try for extreme ranges of motion that may actually damage tissues.
Other dynamic warm up options include leg swings, ankle range of motion movements, trunk rotation that include hip rotation, wrist flexion/ extension/ circles. A Fitness Center Instructor can help you with proper form and technique!