Are you on a well? Barkeeper's Friend (I prefer the powder) is a good choice for stainless. Use a blue sponge - don't use green because it will scratch. I've been cleaning houses for a long time.

Do you have hard water? That may be the reason you have rust spots on a new sink (I lived in a house with the hardest water for miles around for 27 years).

Agree with trying Bar Keepers Friend. Love that stuff.

Using brillo/steel wool will also encourage rust.

I agree with the suggestions to use Barkeepers Friend. Also what is the name of the sink manufacturer? I am not a sink snob, and am fine with the Chinese-made sinks from companies like Kraus, which have proven to be quite good, and made with T-304 stainless steel. But there are some sinks out there that are not very good, and those manufacturers not very upfront about the quality of their stainless steel. Stainless steel that is T-301 has a higher carbon content, and it is the carbon that can rust.

Stainless Steel T-304 vs T-301

Thanks for updating! Looks great!

can I use this clean black granite tile

Stainless Steel a very different material than granite. Barkeepers Friend is too acidic to use on granite tile and its grout. For your tile, just use a bit of mild dish soap like Seventh Generation Free & Clear Dish Soap (a little goes a long way) on a Dobie sponge. Wipe dry with a paper towel or dish towel.