With skill and imagination, you can do a hell of a lot with a voltmeter, a potentiometer and and some LEDs. If you are programming, add printf() statements to that. I built my first computer (6800, 128bytes) that way, and learned a lot.
If you are interested in digital electronics then the prime unique advantage of a scope is that it lets you look at the signal shape and voltage. Such "signal integrity" tests verify that your analogue signals will be correctly interpreted by the digital inputs as digital signals. Ideally look for a 100MHz scope, but a 20MHz scope is better than nothing.
Once you have good signal integrity (which can be a problem on solderless breadboards), you can flip to the digital domain and use a logic analyser and printf() statements.