Old Schwinns are tanks. They last longer than forever.
Likely you just need to take everything apart and clean it. It is possible that you could use new bearings in the bottom bracket and wheels
I wouldn't mess wit the wheels is they don't shake side to side when standing still (i.e. They should spin, obviously, and may wobble due to not being straight and true, but when you sit the bike down and hold it, you should not be able to shake the wheels on the axle.)
Since you will be opening the bottom bracket anyway, I would throw in some new bearings and fresh grease. I don't know the specific model but it looks like you might have three-piece cranks---that is, two arms which hold the pedals at one end and attach to the very short horizontal central bottom tube (often balled the BB (bottom bracket) shell) with a shaft connecting them (usually called the spindle.)
The spindle usually rides on two sets of ball bearings (like fat BBs), one at each end, often in a little metal carrier, or sometimes just in a cup, held with grease. They are usually some standard, readily available size. if the come in a carrier, the carrier is usually ridiculously flimsy and might self-destruct, but you could just pack them in with grease and be fine.
You are right about the tires, they look dry-rotted.
I would replace all the cables and housings as well. if they are that dirty and rusted outside I am sure they are in bad shape inside.
Also check the headset---the bearings on which the handlebars and forks ride. You should be able to spin the bars easily and smoothly. if you feel any notchiness, you probably need a new headset (as little as $10 for a decent one for a threaded fork, last time I bought one.... maybe $20 now, but it will last forever.) Also, the bars should not wobble or shake---they should only spin and not move in any other direction.
Also, decide now if you want to paint it or not .... when it is all disassembled painting is easy. Deciding later means taking it apart again.
That's a heck of a bike. That thing will outlive you even if you are very young. Good find, and enjoy the project.