I have two thermostats in my house that run forced-air gas heat and central air. When I turn on either one to heat or ac, nothing happens. The thermostats seem to function fine when the temperature is adjusted, but the fan does not kick on to blow air out. Any suggestions as to what it might be before I call somebody?

Sounds like a fan relay.

A fan relay? Is that something easily replaced? Any way that I could check it to confirm this is the problem?

Sure.
Locate relay
Then test relay voltage connections with a ohm meter