A very gentle introduction is Boggess & Narcowich, A First Course in Wavelets with Fourier Analysis. I should warn you, though, they're pretty fast and loose with the hypotheses of their theorems. You'll be fine if you've studied advanced linear algebra, and especially fine if you already know some Fourier analysis.
There's also Mallat's A Wavelet Tour of Signal Processing: The Sparse Way. I haven't read very much of it, so I don't have a strong opinion on it yet. It's definitely more difficult than Boggess & Narcowich, but then it probably has about ten times the content (no exaggeration).
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