Not all trainers are created equal. But moving the cg back can make most of the fly better as the novice pilot's skills improve. At the most nose heavy setting, the pitch or climbing is most sensitive to airspeed and throttle setting. This is good if you are looking for self-recovery, but soon most pilots find the ballooning rather annoying.

When the cg is moved back, much of the up trim is removed. This does make flying inverted much easier, and though the dihedral is destabilizing, it is easy to maintain inverted flight with practice, including turns.

My favorite trainer, the Sig LT-40 is even capable of doing outside loops while balanced at the middle of the cg range even with modest power. With the cg moved to the rear of it's range, spins and snap rolls are even possible.

While doing all this practicing, experiment with different props as well. Generally, most trainers respond well with larger diameter and lower pitch props due to their slower airspeeds.

The flight characteristics of trainers can be further changed with modifications to the airframe. Changes to decalage, airfoils, or even different wings can move a trainer from a basic airframe to a more advanced model. But with the explosive growth of RC, a good flying trainer is easily sold if ready to fly.