I think someone has already asked a similar question. I just can't find this thread again. The approach is like this:
-Define specifications of you scope
Bandwidth, Sampling Rate
Input impedance, Attenuator/Amplifier stage
Power supply (dual or single)
Since you want to start with a uC-design I would recommend any of Microchips uC's, since they have a built in ADC, so you can start right away.
Possible answers: Digital (ttl etc) circuit analysis, Analogue (e.g audio signals).
Depending upon which type will define the design strategy, for example if using it for digital signals having only two possible values you need only detect whether the level is above/below a threshold and for how long that state is maintained. THis is simply stored as a few bytes/sample.
For an analoge system you can either successivly sample a waveform and save the data or (a faster method) sample the delta change from the suceeding sample.
A third method involves a mathematical description of the waveform using fast fourier analysis.
Need to know: Analogue or digital?
Max frequency to be analysed.
Suppose you want to look at a 1000 hz sinewave - draw a single cycle out on a sheet of paper, now how accuate do you want it displayed? sketch the minimum number of points then multiply this by 1000 - that's how many samples per second you require. - must go possibly more later..