What are the three evolutionary advances that enabled the vertebrates to survive on land?
The defining physical characteristic of vertebrates, of course, is their backbone, an evolutionary development that dates back some 500 million years and includes a broad array of animals, from birds and amphibians to fish and homo sapiens. With such a vast array of animals that are categorized as vertebrates, finding evolutionary commonalities beyond the development of a skeletal structure is difficult. Because the question is specific to those vertebrates that survive on land, however, there are evolutionary developments that definitely helped them to survive on land.
The first important evolutionary development involved the formation of respiratory systems that enabled early amphibians to survive out of the water. Without the development of lungs, animals would have been unable to breathe the air and could not have survived.
A second important evolutionary step could be the development of hard-shelled eggs, which enabled the reproductive cycle to continue on dry land, with the environmental threats on land differing from those under water. Reptiles, a living link to the distant past, emerged from the seas and evolved according to the requirements of dry land, including hard-shell eggs, as well as skin coatings to protect from land-based threats and to store water, a third evolutionary development.
Snakes are vertebrates, yet lack arms and/or legs, so the following evolutionary development might not suffice for purposes of answering this question, but the growth of limbs was essential for amphibians to emerge from the water and survive on land. Similarly, birds are technically dry land animals, but may not qualify for inclusion in this discussion. Nevertheless, the development of wings and feathers was certainly vital to the survival of that category of vertebrates.
These are just a few examples of evolutionary developments that helped vertebrates to survive on land.