Congratulations on acquiring your Schwinn, and welcome to C&V, tuxfan212.

The first thing you need to do is replace the tires - it is very unpleasant to lose a front tire and have the inner tube wrap itself around your front caliper 16x before catapulting you over the bars while careening down hill.

The second thing is NOT to repaint it. You will negate the vintage appeal/value of the bike if you do.

Next, service the crank, wheel and headset bearings. Grease does not last 40 years and if you run them dry you will damage things.

You can remove rust from chrome with Oxalic Acid (or even vinegar). Aluminum foil is purported to do alright also.

I would not consider spending money "upgrading" the parts on your Varsity if I were you. The parts you have on it may not be so sophisticated, but they are very solid and bulletproof (like the rest of the bike) - That is in large measure the appeal of a Varsity to begin with. The only possible exception would be the wheels - Alloy wheels provide a better ride and much better braking in wet conditions.