Gaining a level is a normal activated ability. It uses the stack and can be responded to.

Each Class starts with only the first of three class abilities. As the first level ability resolves, the Class becomes level 2 and gains the second class ability. As the second level ability resolves, the Class becomes level 3 and gains the third class ability.

Class is a returning enchantment type. Each Class has five abilities. The three in the major sections of its text box are class abilities. Class abilities can be static, activated, or triggered abilities. The other two are level abilities, one activated ability to advance the Class to level 2 and another to advance the Class to level 3.

You can't activate the first level ability of a Class unless that Class is level 1. Similarly, you can't activate the second level ability of a Class unless that Class is level 2.

Gaining a level won't remove abilities that a Class had at a previous level.

You can multiclass or even control multiple Class enchantments of the same class. Each Class permanent tracks its own level separately.