Would it be wrong to think of differentiation as a function whose domain is the set of differentiable functions and co-domain is the set of all functions whose domain and range are some subsets of real numbers?
Yes, differentiation can be thought of as a function from the set of differentiable functions to the set of functions which are derivatives of a differentiable function. (which is, as Dr. MV points out in a comment, not quite the set of integrable functions).
Such things, which map functions to functions, are typically called operators, but this is just a convention, you can think of them as functions just fine.
Differentiation of a function is a linear operator, a function on a set of functions.
Differentiation at a point is a linear functional, a function which maps elements of a vector space to it's underlying field, in this case functions to real numbers.
Look up differential forms. I think you'll find that what you're describing closely resembles them. In fact, any course on differential topology or differential geometry should present this perspective.
Differentiation can indeed be thought of as a function, but it was not viewed that way by the creators of the calculus. The first mathematician to think of differentiation as an operator from functions to functions was Euler's successor Lagrange.