Some antion adults feed on soft bodied insects (for example moths, small flies, aphids), while others feed on pollen. It seems like there are many species who do not feed at all at this stage.
The larvae will prey on any insect (and arachnid) that they can take over. Surprisingly, ants prove to be a poor food item for them in captivity (you need many ants daily to maintain a healthy diet), but other soft bodied insects work just fine - small crickets, silverfish, baby roaches, caterpillars. Beetles adults and larvae are not good.
It is also important to know exactly which Neuropteran family you have. Antlions (Myrmeleontidae) and Owlflies (Ascalaphidae) prefer the above, while lacewings (Chrysopidae) prefer to feed primarily on aphids. Mantisflies are cool looking but have a complex biology unfortunately (larvae feed on spider eggs).