This could go on for pages (Google "bike maintenance" or go to Park Tools website for details), but here are some key things:
Check the tire pressures often, every ride or two. Use a gauge. High pressure road tires need it more often than fatties. Also inspect the tires for thorns, glass etc.
Lube the chain every 50-100 miles, or whenever it squeaks or shifting gets balky. It's worth buying a real chain lube, like ProLink. Motor oil holds dirt and WD40 is too...well, I've used WD40, and it worked OK, but it's supposed to be too light or something. Buy chain lube.
Especially on the kids' bikes, occasionally check the tightness of the bolts and fasteners. Yours, too, but you'll probably notice if something works loose.
That about does it for daily stuff. Eventually you'll have to repack bearings, true wheels and things like that, which requires special tools, but you can pick them up as you need them. Park Tool is a good site for maintenance and repair information, but don't sweat it. Once you get things dialed in, you can ride hundreds of miles with just air and oil.
Oh--CARRY A PATCH KIT OR SPARE TUBES AND A PUMP on all rides. You WILL have flats, guaranteed. And if you don't know how to fix one, google it and practice at home. My first time took half an hour; I can do it now in 3-4 minutes with a frame-fit pump.