If you are interested in the balance between muscle groups, refer to muscle balance.
When deciding how to train to improve your balance, stability, and core strength, specificity is an important principle to consider. For this reason, free weight exercises are preferable to strength training machines. Free weights are better because they work the stabilizer and prime mover muscles whereas machines tend to focus singularly on the prime movers without much stabilizer involvement.
Exercises that heavily involve balance and stability, preferably done with weights (dumbbell or kettlebell), are:
- Split squats.
- Single leg deadlifts.
The following exercises heavily involve balance and power in one movement, and again preferably do them with weights. Aim for a full range of motion with a high degree of knee flexion, i.e., get down low before powering up into the jump:
- Ice skaters.
- Jumping or hopping; forwards, backward, sideways and all directions in-between.
Wanting to improve your balance and stability regarding everyday functional tasks is a common goal for rehabilitation and aging. If this is your goal, taking specificity and safety into account, all of the exercises mentioned above should probably take precedence over unstable surface training.