I own a 10 year old house that has a septic system with a 1,500 gallon tank. Lately I have been hearing numerous “belching” noises coming from various parts of the pipes in the basement. Does this indicate clogging? Does this indicate a serious problem?

First thing I would do is check the septic tank to see if it needs to be pumped. As for your noises, read on.

"Bubbles" and "Gurgles". They are both indicters that something's going on in your drainage system.
A "bubble" indicates a clogged line while a "gurgle" indicates a blocked vent. Let me explain. When you flush the discharge drains down the pipe until it hits a partial clog. Then it rebounds back sending a "bubble" of air ahead of it. When you flush a toilet or drain a bathtub a suction is created by the water rushing down the pipe. This suction is relieved by a open pipe that runs to the outside called a vent. When a vent's stopped up the suction has to relieve itself somewhere.
It sounds like you may have bubbles which would indicate a partal blockage in the main sewer or the septic tank. Good luck, Tom