Is there any good and free Introduction to topological graph theory
My lecture notes on computational topology include some basic material on topological graph theory.  They don't come anywhere close to the completeness of Archeacon's survey (or Giblin's book, Mohar and Thomassen's book, or Gross and Tucker's book), but you may still find them useful.  (Start with "Curves on Surfaces".)

If you're a student, you may also find your local university library useful.  Most mathematics libraries have copies of all the books I mentioned, and they will actually let you borrow them for free, possibly for several months at a stretch.  Sadly, these books may only be available in a legacy format that is difficult to search and copy from, but hey, you get what you pay for.