The exhaust in your car is a system that is prone to attract rust and other malfunctions that will make your car function less well. Exhaust repairs are usually quite expensive depending on how bad the actual damage is. Finding problems with exhaust systems doesn't have to be hard, especially if you're not attempting to perform any actual exhaust repairs. However, just having a look at your exhaust system every now and then can help you get your car to the mechanic's shop before the damage has become significant or even irreparable. You also only need your senses, a proper torch and possibly a screwdriver to do it.
Listen
Strange noises are excellent indicators if something has gone wrong with your exhaust system. Your silencer, for example, is a part of the exhaust system that commonly fails. As it is the part furthest away from the motor, it might have bigger trouble with keeping emissions warm. This can lead to a buildup of acidic fluids that reside in the silencer and corrode it. You will be able to tell something's wrong with it if you hear a roaring sound that hasn't been there before. If you hear knocking or rumbling coming from your exhaust system, this is an indicator that something has broken or come loose.
Look
A visual check might be hard to perform without dissembling the system on the ground. However, you can use a skateboard to roll in under the body of the car. Use a torch to look at the exhaust system and be extra careful to look out for rust. External rust doesn't have to be a sign that you have to go to the mechanics but try to examine all of the items internally as well. If they have rusted internally, then you need to have it repaired. You should also keep an eye on welding seams and make sure that they look stable.
Touch
Using a screwdriver while you're visually examining the exhaust system, you can poke the rusty parts of the system. This will allow you to notice if any part of it seems particularly porous or if it gives unnecessarily much. If you manage to poke a hole in any of the metal, this is a definitive sign you need to have the part switched. If you see any unstable welding seams, you should also try to pull a little bit in the part to see how much it gives. Don't pull too hard as you can actually rip a part of the system off.