Warming up is a breeze...

Warming up is the best injury prevention any active person can perform.  Stretching is not bad, but I would definitely advise against it as a method of 'loosening up' before any physical activity.  I often allude it to taking a rubber band out of a freezer and stretching it right away.  Typically, the consequence of such an action is  that the band will snap and potentially break.  In other words, this is analogous to the functionality of our body's muscles.  Although they won't probably snap as easily as a rubber band, one will not be able to get the most out of their stretching without at the very least some form of a warm up.  The individual's ability to stretch is limited by the constrictive force of their hitherto unused 'cold' muscle.  Therefore, the end result may turn out to be a shorter range of motion during exercise thus cutting short the benefit of an unhindered movement performed in its entirety.

Warm it up before stretching it!

There are several easy methods to warm one's body up in order to gain the benefits of added flexibility during subsequent stretches (which I recommend to be done post workout, not during or before) and prevent injury from coldly performing an exercise which places much strain on a particular muscle group.

 
The most mundane and remedial method of warming up is by briskly walking on a treadmill.  This will automatically enhance the circulatory properties of your blood by bringing in oxygen to your body's highways and roads; i.e., veins and arteries.  Ergo, such flow will eventually work its way to the muscle which will lead to its progressive 'loosening'.  This action leads to better performance during a work out and will prepare the body for its consequent physical activities.  Elliptical machines, stair masters  and all such cardiovascular machines serve to lightly warm the body up.  Granted the aforementioned be performed at low levels of intensity   For one must remember, the aim here is to incrementally warm the body up for injury prevention, not to try and race an imaginary cheetah.

Other methods can be done by performing core exercises to prepare the most important part of the body which plays a supporting role (theater pun intended) throughout all physical activities.  The abdomen and lower portion of the back can be lightly targeted here and returned to a later point with more intense exercises towards the end of the routine.  Never neglect either side for this will cause a problematic imbalance that may lead to future discomfort and possible injury.  Perform light exercises for the rectus abdominis, the erector spinæ and their ancillary muscles to give all ends equal attention.

Core Muscles

Body weight exercises (for the intermediate or advanced lifter) may also server as a wonderful form of warming the body up as most of the movements call for the entire body to act as a cohesive muscular unit.  In this instance, blood and oxygen quickly spreads to all parts triggering this action; i.e., the muscles being utilized.  These can be performed via push-ups, dips, pull-ups, chin-ups, body weight squats, lunges, and others.  Of course a beginning lifter will have difficulty doing this as a warm up for it will initially require much output from the body's physiological faculties that may drain any future energy needed for other exercises that compose the main body (no pun intended) of the routine.

Small Compilation

Emphasis must once again be placed on the purpose of a warm up, it isn't to initially tax the body, but rather to prepare it for the oncoming barrage of physical stress placed on it by strenuous exercise.

Light isolation exercises should not be performed as a warm up as they may actually tax the targeted muscle group too much since the focus is solely placed on it.  Consequently, this specific muscle group may end up being the Achilles's heel for any future compound exercise that involves it in the workout routine.

Lastly, the first exercise of a routine, whether lower body or upper body* is always typically (and rightfully so) a compound movement.  Compound movements are those exercises which recruit a large number of muscles to contemporaneously work together in order to complete the action.  Before performing an active or live set of this first exercise, it is advisable to do a light set as well to get the body accustomed to that motion that will soon be done with more weight.  Not only will this fine tune exercise form as it is extra practice before the real thing, it will also mentally prepare the lifter for the next set and beyond.  The lifter's state of mind is just as important as the body's preparation.  This lighter set should consist of 45-50% of a one rep max and high repetitions in the 15-20 range.  If the one rep max is not known, the lifter may go by an educated guess as to what he believes he could easily handle to serve the above purpose.

*With upper body movements L-Flyes are recommended as a precursor to performing them in order to protect the thin layer of muscle tissue surrounding the rotator cuff by strengthening them in this simple and yet crucial exercise for shoulder health.  L-flyes should never be done with much weight, preferably on the low end around 5lbs-10lbs, never exceeding 20lbs due to the sensitivity of the area being worked. 

So, to recap...

1. Stretching prior to exercising is analogous to taking a rubber band out of the freezer and trying to bend it.  It's not a good idea.  Leave stretching to the end of the exercise routine.

2. A brisk cardiovascular warm up on any of sundry machines such as the treadmill, the elliptical, the stairmaster, and etc. Never exceeding more than ten minutes, and always done at low intensity.

3. Warm the core of the body by performing light exercises with high repetitions for the rectus abdominis, the erector erector spinæ and their ancillary muscles.

4. Bodyweight exercises at higher repetition ranges for the intermediate and advance lifter may also be a great form of warming up.  Beginners should avoid this so as to not tax their body too much prior to the actual workout routine.

5. The first set of the first compound movement should always be done with a very light weight determined by the lifter's calculations or educated guess that can be easily performed for 15-20 repetitions.  This helps with form and prepares the body for the heavier loads.