As a first introduction to the idea of $R[[X]]$, as GilYoung Cheong suggests, just checking the ring axioms from a clear definition of $R[[X]]$ might be an interesting and useful exercise.
Indeed, this provides an example where things generalizing polynomials are not used to define functions in the sense of having point-wise values, but for some other purpose (e.g., generating-functions).
This may cause some cognitive dissonance if one is accustomed to things that either are "numbers" or produce numbers by evaluation.
Another useful, slightly more sophisticated, presentation of $R[[X]]$ which nearly eliminates "checking the axioms" is characterizing it as the _projective_limit_ of the quotient rings $R[X]/X^n$ as $n\rightarrow \infty$. Then the formulas for multiplication in terms of "formal" coefficients are deduced. (I won't write a polemic about projective limits here... partly since such things are widely available on-line, once one knows the keyword.)