There are warm-ups, then there are real warm-ups. If you are going for a fitness run, you're warm-up can be as simple as starting with a slow jog for a half mile then gradually picking up your pace. However, if you are going to do any load bearing exercise, like lifting weights, you need to actually work up some sweat, like you have a slight fever.
In fact, you want to get your core temperature to around 103 degrees F. That's the point at which the collagen and elastin in your muscles and tendons are able to stretch or contract, as they will under resistance, without injury. The heat also improves the ability of fluids to move through and around muscle and connective tissue, as well as the synovial fluid in the moving joints, and thus perform better.
The video below demonstrates an excellent warm-up. What you'll see is not a namby-pamby little static stretch (like you see runners doing standing one one leg and pulling back the foot of the other to stretch the quadriceps) here and there, but an energetic, dynamic sequence in preparation for a more demanding workout.
A routine like this one is a real workout in itself. If you only had ten minutes to exercise, this would be a good one. Add a few more reps of a few of these exercises, like more push-ups, jumping jacks, and squats, and you'd give your heart and muscles some useful exercise even if you didn't do anything else.
Notice how the trainer activates all the major muscle groups and joints. Also, watch how he moves his body in all directions - up, down, laterally, rotating around. He moves without smoothly and without delay from exercise to exercise.
Try the sequence with him. If you can't do it as quickly, do what you can. If you can't do all of them, pick a few of the movements and master them one at a time. When you can do them all, you'll have a really good base of stability, mobility, and core strength to build on.
He's a good benchmark. When you can match him movement for movement and keep your breath under control like he does, you've arrived at a fitness milesone. Then you can ask yourself: What next?