Replace the tires, inner tubes and rim liners. Use modern rim liners, not those soft rubber ones common on old Ten Speeds.
You just might get lots of miles out of an old set of hoop huggers, but then again, you might not. My Peugeot PX10, on its test ride, created a small mushroom cloud when the rear tire blew, and I do mean blew!
In fact, if I would have been clenching a piece of coal between my two gluteus maximus muscles, that piece of coal would have instantly turned into a diamond, at the loud kapow I heard just before my bicycle started to dance a little bit...
If that had happened on the front wheel, I might have crashed the bicycle.
Get the tires or learn how to carry pieces of coal. Best advice I can offer.