A Flying Pigeon is probably the most common bicycle ever. It is a Chinese copy of the Raleigh DL-1 of 1913. It's what all the workers had back when all Chinese bicycled to work.

If the wheels, pedal cranks, or steering fork and handlebars feel loose and can wobble when not turning, the bearings need adjusting. Riding even a short distance can ruin them. For example, lift the back of the bike, grab the rear wheel, and try to shake it from side to side. It should not clunk around. The pedal cranks should just turn and should not feel as if they can wobble around loose, not even a millimeter.

If you get new tires, don't use high pressure. The steel rims are smooth sided on the inside and do not have a hook bead or ridge to grasp the tire bead.

parktool.com or a book from the library ought to have guides on gear cables. The rear derailleur ought to want to be closer to the wheel or in low gear (low to pedal, but the largest sprocket) when the cable is slack. Just attach the cable at the derailleur when the lever is in the slackest position, and make sure the derailleur can move to or just past the outermost, smallest sprocket when the lever is moved to higher cable tension. Then use the derailleur limit screws to keep the derailleur from moving the chain off the sprockets too far to the right or the left.

Make sure the chain's not stretched or you're wearing down your sprockets. The distance between the centers of 25 pins should be 12" and 12-1/8" is too much.