Course outline:
1. Classical Fourier Analysis (Fourier series, Fourier transform on the Euclidean space, discrete Fourier transform and FFT, elements of abstract harmonic analysis on groups).
2. Topics in real variable methods (stationary phase, maximal functions, Hilbert transform and singular integral operators, multipliers, Littlewood-Paley theory).
3. Introduction to some of the modern developments (wavelets, time-frequency analysis, frames).

Recommended Text:
There is no required textbook. However, I recommend the following books:
Part 1. An Introduction to Harmonic Analysis by Y. Katznelson (new expanded Cambridge edition or the earlier Dover edition). 
Part 2. Singular Integrals and Differentiability Properties of Functions by E. Stein.
Part 3. Ten Lectures on Wavelets by I. Daubechies; Wavelets and Operators by Y. Meyer; Foundations of Time-Frequency Analysis by Karlheinz Gröchenig.