The rims on your Schwinn Continental are fairly common 27" size BikePower . There are lots of options available. Continental has both the Gatorskin's and the Ride Tour are pretty flat resistant. I have a pair of Gatorskin's without the extra puncture resistance and I like them.

Get the tires with the standard steel bead. Avoid the type that folds.

You may want to check the sticky link above that highlights 27" tires. I will add that many of us love the Panaracer Pasela's because of the supple ride, however, that comes at the expense of puncture resistance. Despite that I have a pair of them as well.

Another common option that you will see is Kenda's K35. The good points on these tires are that they will look great on your Continental with the tan walls and they are inexpensive. The bad is that they are ok on puncture resistance and the ride is also only ok. I have a pair of these on my Continental and they are fine. No flats so far. If you want I can post a picture of my Continental with the K35's .