This is going to sound really basic, but a good start might be just to reproducibly generate and measure some signals: DC, AC, etc. Make an LED blink. Measure the voltage across the LED as a function of time. Get used to seeing a graph of voltage versus time, and interpreting it. Hook it up to a microphone and talk to it. Measure the headphone output of your PC.
This is all about just learning to think in terms of signals (static and as a function of time) and how they are measured. Try to predict what's going to happen, and see what you get.
It gives the following very basic pieces of equipment for doing hardware development. However this is quite low end and will have lots of limitations. A high end logic analyzer could be $50k for example.
USB Oscilloscope - observe an analog output waveform
This is all about just learning to think in terms of signals (static and as a function of time) and how they are measured. Try to predict what's going to happen, and see what you get.