What is your commute going to be like? Are you simply cruising a few blocks or do you have to travel more than say 5 miles each way? Unless your only doing a few blocks I agree the '60s Schwinn (while overpriced unless pristine) is too heavy and low end for heavy use.
Raising your budget, if possible, will get you a lighter bike suitable for longer commutes and more importantly commutes in bad weather. Unless you commuting more than 5-6 miles look for a decent used mountain or hydrid bike, you can sometimes score a good deal on one.
Lastly where do you live? Lost of members here fix up and resell bikes for fun and little extra money for their own bike habit. Buying from one of these guys will ensure a decent deal and sturdy well functioning bike.
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“One morning you wake up, the girl is gone, the bikes are gone, all that's left behind is a pair of old tires and a tube of tubular glue, all squeezed out"
Sugar "Kane" Kowalczyk