My furnace fan does not always come on automatically. You can hear the ignition and then there is a series of pings and nothing. The furnace is probably ten years old. A Heil model.
The thermostat is a programmable one and when I set the fan to on it comes on fine. Just doesn't always work when the thermostat kicks the furnace on.

Emma, There is usually a switch in your furnace that, in heat mode, delays the blower a few sec's 'till the furnace warms up (so it won't blow cold air). In air conditioning mode the thermostat turns on the blower. It sounds like this switch is sticking or bad. There are different types of these switches (Sequencers, Fan controls), depending on if your furnace is gas, electric.

The good news is that most programmable thermostats can be set to turn on the fan in heat mode. It's either in the program you insert or it may be a small switch you'll find under the front part of the thermostat (the front part of most programmables comes off the wall).
Check your thermostat literature to be safe

There should be something like "FAN ON IN HEAT" in the manual. If no manual, Google your thermostat brand and model for the book. Most T'stat manufacturers supply instructions online.

Good luck

Hope its not a hunter thermostat