What project can I do that involves performing FFT's and signals processing?

Using Arduino Project Guidance
1

I want to get a feel for digital signal processing. Are there sensors I can work with that involve filtering a material?

2

make a tuning device for a music instrument?

door-opener with a "whistle a tune" key?

3

Arduino based audio spectrum analyzers seem to be popular.

I want to get a feel for digital signal processing.

This is not specific to the Arduino but there is a FREE online book:
[u]The Scientist and Engineer's Guide to Digital Signal Processing, By Steven W. Smith, Ph.D.[/u]