If you're familiar with Fourier transforms, I think the bridge between the Fourier worlds and the wavelet worlds is the Gabor transform (a Gaussian-windowed STFT) and the complex Morlet wavelet transform. This is historically how they developed, too. They are basically the same thing, breaking down a signal into "blips" of complex sinusoids:
But the time-frequency space occupied by the blips are spaced differently:
The wavelet version has more frequency resolution at low frequencies and more time resolution at high frequencies, which is usually a good tradeoff (similar to the way the human ear works).
The Morlet is a continuous wavelet, though, so there is overlap/redundancy in the representation, a discrete version is not a minimal representation of the signal, and does not meet the "admissibility condition", which apparently means it cannot be inverted perfectly back into a signal(?), and Parseval's theorem can't be used on it. Modifying the wavelet so these things are possible results in other types of wavelets, and you can eventually work back to things like the Haar wavelet (I think).
Also see What's the difference between the Gabor-Morlet wavelet transform and the constant-Q transform?