A strong understanding of the basics in gymnastics allows more complex skills to be learned more quickly and safely. The stretching and basic tumbling exercises listed below give the body a good warmup and a thorough grounding in the basics of the sport. These exercises, when done correctly, will increase strength, flexibility, and body awareness.

Standing w/ feet together, stretch up on your toes 5 times slowly.
Lift one leg up (try to hold it at 90 or above) and stretch up on your toes 5 times slowly.Repeat for the other leg.
Lift the first leg again and hold it for 2sec (try for 90 or above). W/out letting the leg drop move it to the side and hold and then back into an arabesque. Pattern should be front, side, back, side, front, side, back, side, front, down. Do this for both legs.
Do both Y scales. (Lift one leg straight up to the side and hold it. Make sure to keep both legs straight.)
Standing w/ feet together, stretch down and place your hands flat beside your heels 5 times.

Pike (legs together) press to handstand 3 times.
Straddle (legs apart) and stretch down 5 times. Reach as far back as possible each time.
Straddle press to handstand 3 times.
Do 3 lunges per leg, alternating legs.
Sit down w/ 1 leg straight and the other bent behind (NOT in front). Lean towards the straight leg and stretch down as far as you can. Then stretch down the middle. Finally, lean backwards and lay down (This may take some practice, especially getting back up). Repeat each of these 3 times and then switch legs.

Sit w/ both legs together and straight. Keeping a straight back, alternate pointing and flexing your toes 5 times. Try to keep the heels popped off of the ground by hyper-extending your legs.
Grab your toes (while they're flexed) and pull your torso down toward your legs. Hold for 5-10sec.
Point your toes and pull yourself down again.
L-sit hold for 10-30sec.
2 V-sits. Hold for 5sec each. Pressing through to handstand is optional.

Place your hands on the floor behind you no more than shoulder width apart and stretch down.
Keeping your arms behind you, alternate lifting one leg straight up and then the other 10 times each (20 total). (Single leg lifts)
Raise your arms above your head and lift both legs up 20 times. (Double leg lifts)
Straddle and stretch down both legs 3 times. Hold each for 5sec or so.
Stretch towards the center (pancake).

Stretch out all 3 splits. 2 times each w/ 10-15sec holds.
3 straddle-L presses to handstand. Being able to lower back to the straddle w/ control is just as important as the press up.
3 back-walkovers.

This stretching routine does not stretch the ankles and wrists. They can be done before or after.
When stretching before workout be careful not to do too much static stretching. Stretching any position for more than 20-30sec can lower muscle response time. An example of this would be slower tumbling when trying to punch one skill out of another.
Try to keep good form while stretching. You will get flexible more quickly and also the habit will carry over to the events and form during routines will improve.

Basic Tumbling:
All of these exercises should be done across 1 length of the floor unless otherwise noted.

Forward roll jump.
Forward roll jump full turn. Double turn optional.
Pike (legs together) press to handstand. Forward roll w/ straight legs. Press through is optional.
Straddle (legs apart) press to handstand. Straddle roll w/ straight legs. Stalder press (don't let feet touch) is optional. If you get good at these try press handstand pirouette backward roll to handstand pirouette stalder roll.
Handstand pirouette (turn). Alternate full and double turns. Try to execute half a turn at a time (don't take 30 steps to do a full pirouette).

Front handspring stepout.
Front handspring.
Run and do a handspring punch front handspring (handspring flyspring or handspring bounder). (Only 1)
If you can, run and do a handspring punch front flip. Otherwise repeat the handspring handspring. (Only 1)

Front flip. If you can, do front front. (Only 1)
Cartwheel. Alternate sides.
Side aerial.
Backward roll to handstand. If you can do it w/ straight arms, alternate wide and narrow arms.

Standing back handspring.
Roundoff back handspring. No run.
3 standing back flips.
3 roundoff back handspring back flips. No run.

Some of these skills are more advanced. Do as much as you are able to. The most important thing to think about when doing basic tumbling is execution. Try to do everything technically correct and w/ good form. Otherwise you are wasting your time and might as well skip it.