I have usually heard elements of the dual space being called linear functionals, dual vectors, or covectors rather than 1-forms.
In a smooth manifold (which can be thought of as some space that has a notion of a vector space of tangent vectors at every point -- the tangent spaces):
a vector field is a function that for each point in the space chooses a tangent vector at that point.
a 1-form is a function that for each point in the space chooses a dual vector of the tangent space at that point (the dual space of a tangent space is called a cotangent space, not to be confused with the cotangent from trigonometry, and the individual dual vectors are called cotangent vectors).
A strange way to think of an $n$-dimensional vector space is as the tangent space of a somehow-$n$-dimensional point. Vectors in the vector space correspond to vector fields, and dual vectors are $1$-forms, but I do not believe this is in any way standard thinking.
I think the point of Rudin's definition using $k$-surfaces is that a parameterized surface specifies a list of $k$ tangent vectors at a point, and the differential form takes the point and those $k$ tangent vectors to produce a number. It could be that Rudin is trying to avoid talking about alternating tensors for $k>1$.
I think I first learned how to do calculus with differential forms by reading the beginning of "Differential Forms, a Complement to Vector Calculus" by Weintraub.
There are some good references for more advanced topics at Good introductory book on Calculus on Manifolds