For small simple dainty rings that are of one colour (just silver, just gold, etc.) and ones that come in sets I've found that H&M really fit the bill. They're quite inexpensive, come in three different sizes and most sets include both normal rings and midi rings if you're into that whole hype.
For little hidden gems I'd definitely recommend Topshop. Although they also have sets, I find their single rings to be of better quality and to be a lot more unique. I've been looking at buying single rings from online shops such as Regal Rose and iheardtheyeatcigaretts, but I don't really like buying things when I haven't tried them on. However, I did find a few similar rings at Topshop.
And finally, for big, statement rings I'd recommend Lovisa. It is an Australian brand that we have here in Singapore and it's surprisingly quite affordable and have a ton of unique rings. They also have rings for many different styles, whether you're going for the edgier punk look or more bohemian look.
These were just three of my favourite places to buy my rings for affordable prices. What are your favourite places to buy your rings?