Although my go to book for DSP is always Digital Signal Processing by Oppenheim And Schafer, things have moved on a long way over the last 30 years in terms of books that present DSP in an easier to learn format. The book that I recommend is : Smith, Steven W., “The Scientist and Engineer's Guide to Digital Signal Processing”. The complete book is available for download from : http://www.dspguide.com. I can also thoroughly recommend : Orfanidis, Sophocles J., “Introduction to Signal Processing”, Prentice Hall, Inc. Which is also available for download from : http://eceweb1.rutgers.edu/~orfanidi/intro2sp/.
If you have ever wanted to learn how to write an FFT algorithm (or if you just fancy an intellectual programming challenge, one quiet evening) then I can thoroughly recommend you follow the step by step guide in Steven Smith’s book. Give it a go and see if your solution matches Steven’s, on the next page – no cheating ;-)