Which vertebrate class is considered to be the first that was entirely terrestrial?

Reptilia.

Reptiles were the first animals to develop shelled eggs, meaning they could be laid on land rather than in water as is necessary for the shell-less eggs of amphibians and fish. They also developed dry, scaly skin to prevent the moisture loss amphibians experience through their moist, permeable skins when on land.

The first entirely terrestrial vertebrate class totally independent from the aquatic habitat is the class reptilia.