Borderlands 2, in the skill tree certain skills will be highlighted blue depending on the character's level. It says more points will be granted provided training is completed. What is this training the game is referring to?
Those blue highlights do not depend on character level. They come from a Class Mod you equip.
Before bonus points from class mods take effect on any particular skill in your character's skilltree, you need to invest a point in said skill.
The "training" the game refers to is investing AT LEAST one point into said skill. Otherwise, points from class mods will not apply to that skill.