What is a good resource in which I can read about mollifiers, basic theorems regarding convolutions, smooth approximations of $L^p$ functions and the like? (the presence of exercises would be great, but not necessary)
The course I am taking is very random, there are no given references and the lecture notes are horrible. Any suggestion is appreciated.
"Real Analysis" by Folland would be a possibility.
Also, many books on Fourier analysis/Harmonic Analysis treat this topic (possibly in greater generality than you need), e.g. Folland, "A course in harmonic analysis".
It would help if you added some more details: Are you only considering $\Bbb{R}^d$, or more general topological groups? Are you only interested in approximating in $L^p$, or more generally in the Sobolev spaces $W^{k,p}$?