I am an organic gardener so I rely on natural methods. This is probably what you are looking for if you have children using your yard. Whether you want to use chemicals or organic methods, you will have to keep applying them to have success.
If you can identify separate anthills, try pouring boiling water on them several times a day for a few days, until there is no more sign of ant life. You DO have to be persistent with your efforts, and you might have to do it again in a few months. If they are in your neighbourhood, it won't be long before they are back. This is the nature of the tiny beast!
Scientists don't seem to know why, but one of the things that fireants are attracted to is electrical fields or currents. That means they will often live in or around electrical outlets. Now, I know you can't get rid of these, but it would be a good spot to concentrate your efforts. Of course, it might well be worth your while to disconnect power to outside outlets.
Another idea for control is to mix the contents of two different colonies. This might cause the ants to relocate. But be careful if you try this one!