What type of tub? Cast iron? Porcelain? Fiberglass or acrylic?
Nonie without knowing exactly what kind of tub I focused on the word "old" and found this article that might be helpful
Good luck with your project
First, get a battery operated spin scrubber. I have a hurricane and it speeds the process up tremendously. Then you can use a variety of cleaners but I would start with Lysol Power Scum cleaner in the blue trigger spray bottle. It's gentle but effective. IF you have stubborn stains, try a 50/50 white cleaning vinegar and blue dawn.
My daughter has an older home with stained cast iron tubs. Rust removers helped a little. I scrubbed out some of the stains with Bartender's Friend. Ultimately, I think a product like I used to refinish my kitchen sink is the answer for her.
Here is a video on how to make a miracle bath tub cleaner: https://www.youtube.com/watch?v=1GrCbwZIbwM
For a real dirty old tub use Easy Off No Fume Oven Cleaner in the BLUE can. Works on enamel, porcelain, and fiberglass. Spray it on and let it liquefy. Rinse well.
Old bathtubs to me, mean enameled or porcelain over steel/iron. Is that what you have? Or old plastic tub?
Just like our teeth enamel, that enamel wears down over the years.
I would use a bleaching Ajax or Comet on it and polish it in a swirling motion.
This might help: https://www.hgtv.ca/bathrooms/photos/how-to-clean-a-dirty-bathtub-1925086/
use a bathroom cleaner with bleach