What you'll need in order to understand ant to know DSP is Discrete math, Fourier Transformation (fot the DFT, FFT etc.. stuff) and of course, the foundemantal math which include trigo. and a little bit calculus math.
as mentioned before, you''l need some Exponential understanding (when you will leran the Z-Transform).
About the Calculus math, It'll be needed if you gonna 'upgrade' your self from DSP to PSP (Picture Signal Processing), because there you must use multiple integral (double integral form in the FT) in order to make any picture processing.
Now for the punch line: When I got my first DSP lesson I held in my hand the Proakis Book : Digital Signal Processing ---- try to get it; its very good book which covers all the basics and continue to the advenced topics.
I think it's gonna be hard for you so I recommend you to get, some how, the following book:
Digital Filtering: An Introduction by Edward P. Cunnigham.
This excellent book covers in the first two chapters, in very clearly way, the basics of the DSP. (After the 2 chapters there is some theroy of filter in an analog manner which cont. to the digital design - its not relavent for you, in your case, but its worth to see)
Bye !
Itzik