I'm looking for some advice on the best way to treat / repair some rusting on my kitchen sink . I've included some pictures below. I'm not sure what material the sink is (is there a way to tell?).
It's an enamel coated steel (verify with a strong magnet). There are DIY repair kits, but your rust looks pretty serious. You'd need to start by taking the sink out, using a grinding wheel to get down to bare steel, then building up epoxy/enamel repair.
Yours may be a builder special of low quality: and the rust looks pretty bad. I'd recommend a new sink. Recycle the old one as steel scrap. And caulk the new one well to avoid water pooling under the edges.