Basic equipment for getting started in digital electronics/microprocessors
A digital storage oscilloscope. Preferably with at least 50 MHz bandwidth and preferably with at least two channels. There are many low-cost options out there, either stand-alone or PC (USB)-based. There's also a lot of servicable used gear on places like ebay.

With this one instrument, you'll be able to measure voltages, verify that oscillators are working, and view more complex time-based behavior that's too fast to see on an LED. Plus, if it has PC connectivity, you'll be able to capture waveforms and share them with people who can help you solve your problems.

I see too many people who start complex microcontroller-based projects that get stuck on an issue and end up very frustrated without this one essential piece of equipment.

I think the second item would be a decent bench power supply. 0-20V, 0-2A at a minimum. A lot of basic start-up issues can be skipped if you don't have to guess about the quality of the power going to your circuit.

Anything else is going to depend on the types of applications you want to get into, and the specific technologies you intend to use.