A standard reference for this is Hirsch's Differential Topology textbook. If $X$ is compact near all the topologies you'd like to consider are essentially the same. Sometimes they're called the Whitney topologies, or the $C^\infty$-topology, there are weak and strong variants that are only relevant when $X$ is non-compact.
These spaces have the homotopy-type of the space of continuous maps. The basic idea is to consider $Y$ as a submanifold of some Euclidean space, any continuous map $f : X \to Y$ you can apply a smoothing operator to, then project via the tubular neighbourhood theorem to get a smooth map $X \to Y$ approximating $f$ (in the $C^k$ sense for any $k$ as large as makes sense for your given map $f$), similarly you can apply this construction to families of functions.
Hirsch doesn't bother to get into the details of the Frechet manifold structure on these mapping spaces but it's available. Kriegl and Michor's "Convienient setting for global analysis" is a fairly comprehensive (if daunting) reference for this. But there are other references out there that provide a modest amount of details.
http://www.ams.org/publications/online-books/surv53-index
We have one of the founders of this subject online -- Richard Palais. Perhaps he will have some comments eventually.