What general equipment is needed to 'explore' circuits?
I can think of a small handful things you might need, depending on what you're doing:


An oscilloscope - essential if you want to examine waveforms and compare voltage vs time.  Needed for timing circuits, clocks, oscillators, or anything of that nature.  Could also be needed if you're doing RF stuff, but scopes that can measure at the higher RF frequencies get crazy expensive quick.  But a decent 100mhz scope with a 1G/s sampling rate can be had for &lt; $400.00.
A frequency counter - May be slightly redundant with the oscilloscope, but a good standalone frequency counter can be a very useful tool. Again, this mainly needed when you're working with any kind of timing/clock signal or anything emitting a constant frequency.
A function generator - so you can deliver "known good" inputs to a circuit and then compare the output with what's expected.  Say you have a circuit that expects a square wave at 5hz as the input, but it's not behaving.  You think your input is good, but want a way to sanity check. You can feed your circuit the expected input using a function generator and see if it works. If so, you know your input is off somehow. If not, the problem is elsewhere.
A logic analyzer - this can be pretty inexpensive for a basic analyzer.  A Bus Pirate will work, or you can just run software on a Beaglebone Black and make it a Logic Analyzer, optionally with a cape that does level conversion if needed.  There are dedicated benchtop logic analyzers, but they tend to be expensive and more specialized.  Some oscilloscopes also have optional add-ons that let them double up as a logic analyzer as well.
The other "main" piece of test equipment I can think of is a spectrum analyzer, but I would consider that much more specialized and not something you're likely to need for basic circuit exploration.


Edit:


A high quality lab power supply with current limiting.  Current limiting because sometimes you aren't sure your circuit is right and you don't want to risk accidentally dumping a gazillion amps into it.  Up to 12 volts covers probably the bulk of use cases a hobbyist will run into, but if you're getting a quality lab power supply, it'll probably do 30V or better anyway.  


That said, if a proper lab power supply isn't available and you can get by without current limiting, you can get a lot of mileage out of a generic ATX PC power supply.  Those give you 12V, -12V, 5V, and 3.3V out of the box.  I use one with a homemade circuit board with an LM317 adjustable voltage regulator for times when I need 9V or 6V or something else.