Have you tried the CLR liquid? It removes mineral deposits (calcium, lime rust) from fixtures and glass shower doors. It is harsh, so use gloves and use fan or open windows. You can usually find smaller containers in the supermarket cleaning isle.

https://clrbrands.com/Products/CLR-Household/CLR-Calcium-Lime-Rust-Remover

Now you need to figure out if the previous owners were just slobs or you have really hard water and should look into a water softener.

I’ve had good results with BioClean on my chrome faucets, shower fixtures. I get it on Amazon.

It appears as though some scratches have been made on this chrome. If you know the brand of faucet, I'd look at their website for cleaning instructions. If you have very hard water, you'll most likely need a water softening system, but you may already know that chrome fixtures require drying after use to avoid water spotting.

Maybe this a vocabulary issue, but those are not stains.

A combination of mechanical damage from abrasives and hard water deposits is my guess from you pictures.