For an old bike like that, you can get a pretty good finish with ordinary hardware store rattle cans, like Krylon. I've painted several old frames that way. The secret is in the preparation--you need to get off ALL the dirt and grease, then sand lightly all over, even in the tiniest crevices. It's much easier if you take the bike apart, removing brakes and everything, first. You can use a paint stripper if you want, but it's really not necessary to go down to bare metal if you get the surface clean.
Once that's done, wipe the surface with a tack cloth, mask anything you don't want to paint, then spray. I always prime first to even out the color, following directions on the cans, then apply two or three or four light coats of paint with a light sanding between them. It will look good, but won't be nearly as durable as the factory paint.
You might check for a powder coater near you, too. They do all the work, including the prep, and the finish is super-durable. Haven't checked prices lately; five years ago I paid $60 for a kid's bike.