New riders tend to ride in the chaff on the side of the road. Seasoned riders know that is where the thorns, bits of glass, shards of metal, and other flat-causing things live and they naturally ride just left of that chaff. Be mindful of this and you'll have far fewer flats.

Also, if you ever do find yourself riding through a bit of broken glass or something, you can probably pre-empt a flat by carefully sweeping your tire with a gloved palm of your hand (both front and rear) and this will clear the debris before it can make its way through your tire to your tube.

Keep your tires up to correct pressure to eliminate pinch-flats by inflating every day you ride. Tires that need this high pressure and that are so thin will normally lose pressure enough in a day or two to be a big difference. Just always pump before you ride.

I would avoid slime or any of that gooey crap as it makes a huge mess when the tube inevitably does fail and it is heavy and can gather in one section of tube, causing a balance problem.

When you buy new tires, make sure you get the ones that have some kind of flat protection lining (almost all high end tires do these days) and when you buy tubes, don't get lightweight race tubes, get the slightly thicker ones that will hold air longer and resist flats better.

Enjoy your new bikes!