Which new brake set? There are plenty of different models out there. To get ones that'll work for you, you really need to measure reach on both front and rear to make sure your new brakes are sized appropriately for your frame/fork/wheel combo. If you don't already have your 'new' 700c wheels and you have 27" wheels instead, measure reach on your 27's and add 4 millimeters.

With that said, assuming your new brakes have a reach that'll work for you, they'll probably stop the bike very well. Just about any modern dual-pivot sidepull caliper from Shimano is a decent brake and easy to set up. Some of the newest model high-end brakes have a different cable pull ratio and might not work so well with old brake levers, but if you're not going out of the way to buy high end parts, you'll probably be fine.

I'll assume the "old style" brakes "rigged to your seat post" are centerpulls. Maybe Weinmann. Decent centerpulls can work pretty darn well. So long as they're adjusted well and the brake pads haven't hardened too much with age, they'll stop you. (Maybe not too quickly if you've got steel rims, but that's a problem with the rim, not the brake.)

Something else to be aware of: Caliper brakes are generally available in two mounting types: "nutted" and "recessed". Your Schwinn's original brakes were almost certainly mounted in the classic "nutted" style. The mounting bolt from the caliper sticks all the way through the frame or fork, where it's held in place with an exposed nut. Most brakes sold nowadays are the newer "recessed" style. The mounting bolt ends in the middle of the fork or the frame's brake bridge, hidden from view. A special recessed nut reaches into the frame or fork from the backside and the two are threaded together inside the bike. You'll want to make sure the brakes you buy are the nutted variety; otherwise you'll have to get creative to mount 'em. Sheldon Brown's web site has more info on both brake types and mounting options.