An oscilloscope is the best way to observe active signals and if they are toggling correctly or look like they should - you can compare the trace on a bad board to a good board. You can also measure voltage with it. Also great for debugging power supplies. They are your eyes in a circuit. You don't need to understand all of the functions of the oscilloscope to use one as a beginner, just basic functions to set beam position, voltage range, and to adjust for a steady signal. You could remove over 2/3 of the controls and it would still be useful for your purposes. I bought my HP 100MHz scope for $25 on craigslist. The clip with all of the leds to show the logic states of all pins is also very handy. I can clip it on the control IC on the board then observe the controlled signals with my scope.
If I could only afford one test instrument, it would be the oscilloscope due to it's versatility. The other would be a multimeter, one that also measures frequency and capacitance and audio for continuity. I bought a talking one that has proven very handy - I can focus on placing the probes on the correct pads and hear the readings.
Logic probes with audio are the way to go - again that audio lets you train your eyes elsewhere.
But nothing beats an oscilloscope..... Like a desoldering station, you'll wonder how you ever managed without it.
Bill