What could cause a foghorn sound whenever the toilet is flushed or the faucet is used?

Unless you live near a major seaport and have been victimized by a cruel coincidence, I'd say it is a common phenomenon known as "water hammer" and easily fixed. Water hammer is from ordinary vibrations in copper pipe supply lines. It is never noticed when the piping is properly braced and isolated. Check with a plumber or your hardware store expert if you think you can do it yourself. Ask about "hammer arrestors" as well.