As a child a friend of mine would often have his one eye covered to make sure he did not rely on only one. Observing human mechanics one can deduce an individual's favored side. Even ambidextrous individuals have a favored side. When training the complimentary side, there is a great effect of bilateral transfer to the dominant side; however, the dominant to compliment does not have the same affect.
So, skills need to be practiced on both sides. I do this myself for various sport specific movements, and it can be done to help to gain strength while preventing excessive loads. Additionally, most functional movements are done in an alternating unilateral fashion. Running is a series of bounds. Upper body functional movements like punching is nothing more than a single arm push. A block in basketball is a single arm raised. Kicking in general requires stability with very little support.
Exercises should be progressive. Try going from bilateral to unilateral supported. Next alter the support limb's stance to decrease external aide. Finally, perform the unsupported unilateral activity.
Overhead Press/Side Press/Military Press (Dumbbell/Kettlebell to Barbell)
One of the few exercises that does not need to follow this progression protocol. It is actually easier to teach a correct single arm version of these exercises than a standard military press. I would recommend starting with single arm overhead presses before starting any inverted stand.
Single Arm Push Up or Bench/Floor Press
The push up is an easier variation to progress and modify to make more difficult. Additionally it takes a tremendous amount of core stability to perform. Starting with a staggered push up (one wide, the other close) you can begin to put more pressure on your arm. Adding an unstable surface such as a basketball to the supporting arm will really emphasize the other arm as you struggle to maintain stability. Keeping the arm straight and rolling it across the floor or the basketball (Lever Push ups) will cause an even greater amount of force to be exerted on the arm. Finally the 1 arm pushup can be acheived by changing the angle at which you perform it. A single arm wall pushup should be practiced when you are able to perform a standard pushup. Progess the angle until strong enough to do it. Doing a single arm divebomber or a single leg single arm pushup with or without evelvating the feet can add to the level of difficulty of this exercise. Finally, you can add weight to make it even harder.
Squat, Split Squat, Elevated Split Squat, Pistol
Starting with a good deep squat is pivotal for mastering these exercises. If you are unable to squat deeply, then that is the first step. This can be helped with split squatting (lunges) which will help strengthen the core, enhance balance, and prepare the legs for a greater load. Doing an elevated split squat (or Bulgarian split squat) will force even more weight on the front leg and encourage greater levels of balance. This can 
be done with weight or with a suspension training system or even just a bench/box. Finally, the pistol. The ultimate in functional squatting strength. This will allow you to load your quads and knee dominant lower extremity patterns efficiently without putting a lot of pressure on the spine. Additionally this is a great rehabilitation tool to help with ankle and hip balance strategies.Deadlift, Lunge, Suitcase Deadlift, and the Single arm Single Leg Deadlift
The deadlift is all about hip dominant strategies. Lunging will add that level of stability if paired with loading at the sides instead of in a front/overhead position. Moving to the suitcase deadlift will allow you to feet the weight differences generated by weight being loaded on only one arm. Finally hitting the single arm single leg deadlift paired with the pistol will create superior stability joint integrity and control. This will prevent injury and give the individual a proprioceptive edge over competition. Finally, it allows the pairing of core and hip activity to create a functional unit of POWER!
Bilateral Protocol 3-5x3-5 (Novice, Beginner, Intermediate, Advanced, Elite)
Program A
Squat, Split-Squat, Elevated Split-Squat, Pistol, Loaded Pistol
Kneeling Push, Standard Push,Staggered/Lever Push, SA Push, Loaded/Elevated Push-Up
Program B
Deadlift (DL), reverse lunge DL, Suitcase DL, SL DL, SA SL DL
OH Press (DB), OH Press (BB), OH Clean & Jerk(DB), OH Snatch (DB), OH Snatch (BB)
