Whiteflies look like tiny white moths and live mostly on the underside of plants' leaves. The larvae feed on the plant sap and debilitate it and can also carry disease. The adults are attracted to yellow sticky cards; freshly caught adults are white and dead ones turn into little dark tan specks.
Fungus gnats/shore flies look like tiny little house flies. They fly up from the surface of the soil and deposit fecal specks all over the tops of the plug trays. The larvae live in the soil and feed on algae growing there; there have been some who say that they will also damage tender roots. The adults are attracted not only to the yellow cards, but better to light blue ones.
Whiteflies like warm, dry conditions. Insecticidal soap works well for us (use a specific insecticidal one, not a dishwashing detergent one which can burn seedlings). Since it is a contact killer, you will have to make sure the spray contacts all the underleaf area to get the larvae. There are other chemicals which work, and once you have identified which insect you have, anyone who sells the products will be able to help you.
Since the algae is the root of the problem with fungus gnats, it's best to get rid of that (the adults only live about 3 days and do not harm the plants -- but they're a nuisance). Let the seedlings dry out a little between waterings if possible, but don't let them wilt. This is kind of hard to judge with the small root ball of a plug, but it's possible. We have used parasitic nematodes in the water to kill the larvae; we now use bromine in the water to prevent algae growth. The disinfectant Greenshield also works well, but isn't labelled for on-the-plant use (but we've done it anyway . . .). There are also chemicals that you can drench with to kill the larvae, and sprays to get the adults.
Best of luck! Insects on plant babies are no fun.
Marcia in MT (not logged in)