The most important principle of 20th-century physics is that all observable properties of things are about relationships. Even space and time must be spoken about in terms of relationships. There is no such thing as space independent of that which exists in it and no such thing as time apart from change. This is an old idea that philosophers such as Leibniz have argued for centuries, but general relativity is the first physical theory to be based on it. Many of the important problems facing theoretical physics have to do with the change from an absolute view of properties to this relational view. For example, I think it is highly likely that this is the key to further progress in string theory.
A very important part of turning cosmology into a science is to understand all the implications of a seemingly trivial statement: There is nothing outside the universe. One aspect of this is that there can be no observer outside the universe. We must understand the universe in a way in which the scientific description of it is a description made and used by observers who are part of the system itself. This seems to go against the idea that the scientific view of nature is objective, and an objective description is always based on observations of a system from outside. If cosmology is to be a science, we must invent a new notion of objectivity that allows the observers of the system also to be part of it.