Most general-level physics class teach DC circuits with resistors. Equipment for this unit is easy to find and use -- Radio Shack, Harbor Freight, or any online electronics wholesaler can sell you meters, resistors, breadboards, light bulbs, and power supplies.
Some advanced courses, including AP physics B, teach DC circuits with capacitors, as well as the "after a long time" behavior of an RC [resistor-capacitor] circuit. The best demonstration tool I've found for this unit is the 1 farad capacitors available from science supply stores. Why 1 F, which is prohibitively large for any useful application? Simply because the class can do quick Q=CV calculations in their heads during my demonstrations. If I have students work with capacitors in a laboratory setting, I buy some cheap capacitors from Radio Shack.
Some very advanced courses aimed at future scientists, including AP physics C, teach DC circuits with resistors, capacitors, and inductors, including the time dependant behavior. RC and RL [the "L" indicates the presence of an inductor] circuits reach a maximum or minimum voltage exponentially; the voltage of an LC or RLC circuit oscillates. Teaching the mathematics of exponential decay and the meaning of the "time constant" is a tough challenge. But so is demonstrating the behavior of these circuits.
Showing the time-dependent behavior of an RC circuit can be straightforward -- hook the 1 F capacitor in series with a battery and a variable resistor box. The time constant (equal to RC) can be varied by adjusting the value of the resistor. Measure the voltage across the capacitor or resistor with a Vernier voltage probe and Logger Pro software. You can show the variation of voltage as a function of time, which generally matches theory quite precisely. Add a Vernier current probe, and you can make even more graphs.
When I started teaching the behavior of an inductor in a circuit, I gave up on cheap equipment; I just bought a PASCO RLC demonstration device, shown in the picture above. This card has easy connections to several resistors, several capacitors... and an 8 mH inductor! Now, an 8 mH inductor gives RL circuit time constants on the order of milliseconds. But that's okay -- the voltage probe can be set to take data 1000 or even 10000 times per second, as long as you only take data for a second or two. You zoom in on the voltage-vs.-time graph, and voila, you see the correct exponential behavior.