To my mind, the classical subject is quite different from the modern, evolved form of the subject
I started on the classical side with Yitzhak Katznelson's An Introductio Introduction to Harmonic Analysis: This is in the classical camp: Lots on Fourier Series. Very clear; very nice proofs. You will learn lots of gems about trigonometric series. In this classical camp, Zygmund's treatise Trigonometric Series (two volumes) deserves a mention. This is also a very beautiful book.
For 'harmonic analysis' as a modern field, you ought to get your hands on a copy of Stein's books as in Peter's answer. The late Tom Wolff has a very useful set of notes in this regard, available (I think, still) from Izabella Laba's homepage.
I also second the recommendation to look at Tao's old dvi/pdf notes on his website and later on on his blog. For example, I remember finding his post on interpolating $L^p$ spaces very nice.