I think that if you get the stock wheels checked out to make sure they are true win proper spoke tension, you'll be fine with some 27 by 1-1/4 tires. I like the Forte Strada Kevlar from performance bike. They are usually on sale for 12.99. Continental and Vittoria also make a decent 27" tire if you are feeling a bit spendier.

You can't run 700c tires on your 27" rims. 700c rims are smaller in diameter by about 8mm. If you are looking to run 700c wheels in your bike, the brake calipers will need ling enough brake pad slots to move the pads down another 4mm (this is th difference of radius between 27" and 700c rims). Since, you bought this bike to see how you like road bikes, it is probably best not to spend too much money on the bike if you don't have to. If you like it, you can probably recoup most your money to spend towards a new(er) bike or keep the Schwinn as a backup.