How to check if (small) aircraft's radio is working properly?
Here are some things you can do, there are probably more. I'm assuming you're mainly interested in checking COM, not NAV.

Controlled airport

Check if you can hear the ATIS, Ground or Tower; if you can, then at least you know you can receive. But you might as well just call Ground immediately and ask for a radio check. The correct phraseology in the US (see the Pilot/Controller Glossary) is "how do you hear me?":


  Bowman Ground, N12345, radio check please. How do you hear me?


Uncontrolled airport

Listen to the AWOS, if there is one, to determine if you can receive. Ask UNICOM or another aircraft to give you a radio check. If none of those options are available then you can use a handheld airband transceiver to test. Tune it to the frequency you're listening on and click transmit a few times, as if you're turning on pilot-controlled lights (PCL). If you hear the clicks in your headset, you can receive. To test transmission, do the opposite: click the transmit button on the yoke and listen on the handheld.

Airborne

You could call Approach, Center or FSS; try activating the nearest PCL (works best at night); tune and identify a navaid (that doesn't verify COM works, but at least you know something's working); listen to 121.5 (in the US, you should be doing that anyway) because there's usually some bored pilot asking for sports scores on it and some other equally bored pilot yelling at him to "stop talking on guard!".

Troubleshooting

As always, if you have any problems with transmitting or receiving do some basic troubleshooting first: check you have the correct radio selected; check you have the correct frequency; check the volume controls on your headset and/or panel; turn the squelch up/down/off; check your headset is plugged in correctly; if you have a right-seat passenger, see if they can hear/transmit; and anything else you can think of.