Do not touch the camera’s shutter. In most cameras it’s very very delicate. Even a blast of compressed air could screw it up.
If the controls all move smoothly and accurately with no play or grinding, you should be ok. Get some lens cleaner from an eyeglass shop. Never squirt the cleaner directly onto a lens or camera body! Apply it to a lens cleaning cloth (no paper products!) or q-tip. A can of compressed air or a blower brush will get dust out of the nooks and crannies.
If the camera is pretty dirty or dusty, it might be best to have it professionally cleaned. I’ve had several ancient camera bodies done for <$100 each. The came back immaculate and in tip-top condition.
To expand on Padeye’s camera test:
Get a roll of 100 ASA slide film. This will just be a test roll, so you don’t need a slide projector or anything. Use slide because what you want to test will not be visible in prints. (This is because any over- or under-exposure on the negative is corrected when the print is made.) You could just look at the negatives, but it’s easier with slides. And I like slides. Everyone should use slides.
On a clear sunny day, point the camera at the sky, but well away from the sun. No ground, no telephone poles, no clouds, just blue sky. Take a series of exposures at the following settings:
1/2000 at f4
1/1000 at f5.6
1/500 at f8
1/250 at f11
1/125 at f16
1/60 at f22
Take note of which exposure number had what exposure. This will test if your camera’s exposure settings are consistent. When you get your film back from the processor, compare your blue sky exposures to each other by holding them up together in front of a window. They should all have basically the same color blue. A tiny bit of variation is ok, though.
Your camera probably has some kind of exposure meter. It’s probably a needle or a series of tiny lights in the viewfinder. Checking the accuracy of that is a little tougher. As Padeye said, a medium tone in bright sunlight should meter f16 at 1/125 using 100 ASA film. A gray card is best, but grass is pretty close to medium.
You could also borrow a friend’s camera (which has an exposure meter, of course) and compare the reading that each camera takes of the same subject at the same time. Make sure that neither lens has a filter on it, both cameras are set for the same film speed, and no zoom lenses are mounted (most zoom lenses shift the effective aperture as they are zoomed. The camera’s meter accounts for this, making it not a problem in normal use but it will screw up the comparison test.) Both cameras should give the same reading, within 1/3 of a stop or so.