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 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][1]. 

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.


  [1]: http://www.math.ubc.ca/~ilaba/wolff/