How do I paint on a old lamp?

Spray paint works well, especially if there are odd shapes.

You can clean , then paint with chalk paint. Be sure to add the clear top coat.

If metal....use a green kitchen scrubby to dull the surface. Blue painters tape and tape off what you don't want painted. Several light coats with spray paint for metal. If porcelain...just clean and dry well. Use Rustoleum 2X spray paint. Several light coats. Will give you a smooth finish.

You can spray paint it or chalk paint it. It really is up to you. Here are links to two pretty lamps I made over using chalk paint- https://www.justalittlecreativity.com/2014/01/simplest-lamp-makeover-ever-using-chalk.html

https://www.justalittlecreativity.com/2018/11/farmhouse-style-on-a-budget.html

I agree that either spray paint or chalk paint will work. Have fun!