What are some good Fourier analysis books? I have taken real analysis, but never learned Fourier analysis. What is a good book to get started? I'm not sure the Stein book would be good.
"Fourier analysis" by T.W. Körner and its accompanied exercise book are the best.
please don't propose books that are impossible to find on internet, there are enough books and courses that are free to read on internet, 

and the scientifics that you are should understand it is a huge social progress that everyone can FREELY  access to the scientific knowledge.

look at any of the first links for :

google / fourier transform pdf

google / discrete fourier transform pdf

google / distribution fourier transform pdf

google / laplace transform pdf

google / filter convolution pdf

google / linear differential equation pdf

but be careful, it can take ~ 5 to 10 years to fully study the subject, it is so vast