While all the books listed in other answers are recommended books, neverthless they are not specific for filters and skip basics. The main issue when learning digital filters is that most learners are uncomfortable moving from time domain to frequency domain. Then there is math one needs to deal with. Add to this lot of jargon specific to filters, this makes learning filters a very non-intuitive and hard exercise.
Before starting to learn digital filters, I think one needs to be at least comfortable with basic concepts and jargon of Analog Passive(RLC) Filters such as Impedance, Phasors, RMS, Frequency sweep, dB scale(octave/decade), Transfer function, filter order, Gain, Attenuation, Magnitude & Phase Frequency response(Bode Plots), -3dB(cut-off) frequency, roll-off, Passband/stopband, ripple, Bandwidth, resonance frequency, Quality Q factor(sharpness), tuning, s-domain(=jw), domain transformations(time-domain to s-domain to z-domain), Laplace transform, z-transform, Types of filters (Low pass, High pass, Bandpass, Bandstop, All pass), Basic 1st order/2nd order filters, applications of basic filters(audio, radio, anti-aliasing, etc.), designing filters for given specifications, filter stability analysis(pole-zero plot) etc.
In order understand some of the above concepts and jargon, the book I will suggest to read (in order are as follows:
- The Art of Electronics by Paul Horowitz and Winfield Hill: sections 1.3-1-5.1, 1.7, 6.1-6.2 (50 pages) and Learning the Art of Electronics by Thomas Hayes and Paul Horowitz: sections 2N, 2L(lab), 2S, 2W, 3N.1-3N.4 (70 pages)
- A Basic Introduction to Filters — Active, Passive, and Switched-Capacitor by Kerry Lacanette (24 Pages)
- (Optional) Design and Analysis of Analog Filters: A Signal Processing Perspective - Chapters 1 and 2 (100 pages)
Once the above concepts are clear, you will gain an intuitive understanding of filter design. There after you can pick any of the recommended digital filter design books and I assure you that most of it will be a cakewalk. In any case, here is a book I recommend for making transition from passive analog to digital filters:
- Digital Filters for Everyone by Rusty Allred