You don't say what your oscilloscope is, so I'm assuming it's the Tek 2465. It measures voltage. Positive voltage makes the beam go up, negative makes it go down. Play with a 1.5v battery and the input mode set to "DC Coupling" to get the idea. It has a very high input impedance (1 megaohm) so it won't load the circuit you're probing much. It also has a very low 50 ohm input impedance mode: for now, don't use this mode because it's pretty easy to burn out. That's for later.
The ground lead on the probe you buy is directly wired to the ground (3rd prong) on the oscilloscope's power plug. This is where it's fundamentally different from a multimeter in the blowing-stuff-up category. For example, if you connect the ground lead on the probe to a plug socket's hot line, you would short it to ground. A multimeter is isolated and floating, so connecting the black lead to a plug socket's hot would not short it out. Always ask yourself if it's safe to short what you're connecting the ground lead to to ground. Also keep in mind that all the other channels' grounds are connected together.
Whether you can probe a plug socket's hot depends on the probe you buy. The 2465 allows up to 400V on it's input, but it may not be safe to do so with your probe. Learn about frequency derating with your probe before attempting any higher voltage probing. I will say that probing the mains is almost always a bad idea with no real need: you can use a transformer and see almost exactly what's going on with that.
The only current that passes through the oscilloscope is what the 1 meg input impedance (or 10 meg with a common 10x probe) will allow. (This is a simplification: there is capacitive reactance too which you get into with that derating thing above. Simply put, at very high frequencies your probe stops looking like 1M or 10M resistor and rather starts looking like a very low resistance to ground.)
Checking a (low voltage) power supply for cleanliness is certainly something that your oscilloscope is quite capable of. The best mode for this is with an "AC input coupling" which blocks DC so you can amplify only the AC ripple component of the power supply. Suppose you wanted to look at 100mV rippling on top of a 12V supply. In DC mode you'd have to set it for 2V/division since you only have 8 divisions to keep the trace on the screen. Your ripple would be tiny and hard to see since it would occupy only 0.1V/2V = 0.05 of a division! In AC mode the 12V DC is blocked and you can go into 100mV/division or even lower.