DSP is a huge subject, and can get really quite gnarly in terms of maths. However, many guitar effects are quite basic in their function. Tremolo for example simply modulates the input signal with an oscillating signal (sine, triangle wave etc..). That just requires a table for the modulating wave, and multiplying the input with it. "Distortion" is a subject in itself, but essentially is just clipping the signal peaks which introduces higher harmonics. Delay is just how it sounds, the signal is passed through a delay line (= RAM, lots of RAM..) and fed back into the input after attenuation. Phasers, literally mixes the input signal with a variable phased version of that signal, and chorus uses slight delay and pitch changes.
There was a chip specifically for easing one into DSP effects, the FV-1 by Spin semiconductor. I have one, but never got around to playing with it. It was rather basic but had everything one needs to get started on chip, ADC, DAC, RAM buffers etc.. It didn't have the best support, but there were libraries one could use as templates.
Filters are a subject in themselves but are essentially achieved with a series of multiply and accumulate (which is why DSP specific devices have dedicated hardware for this function, and can do many in parallel, and why the teensy with an M4F was mentioned). The trouble with a lot of educational sources is that they tell you what is done in DSP, but not necessarily why or even how. I'll have to search for an excellent introduction to DSP filters I have bookmarked somewhere. It starts off with your standard window averaging, and eventually gets to IIR vs FIR and how to implement them, and it quickly becomes clear why DSP chips are the way they are. Can't find it right now..