I just moved into a new home and noticed that each time the A/C comes on it starts up for about 3 seconds, shuts off, then starts up again. This second start-up is when it stays on until the desired temperature is reached. I have a Honeywell SuperStat thermostat that is about four years old and the A/C unit is about 13 years old. Is this normal?

No this is not normal. We call it short-cycling and it can cause damage to the system.

Sounds like a defective or out of range low/high pressure control. As MarkwithaK said this can cause damage so have your unit serviced right away.

As hvac1000 said, get it inspected by a local service company. I highly doubt that you have a low pressure control as they are not that common on residential systems but it does happen.

A run capacitor can also cause this problem. Unit will start but not run. Check wiring to capacitor and compressor in outside unit.

With that old of a system it could also be a weak valve putting a larger load than normal on the system causing the internal overload to open. Could be a weak overload... just way to many options to diagnose without physically being there. Save yourself a ton of headache and call your service company.