NetNewsWire is great on iOS and OS X. Use the share sheet to subscribe to blogs.

I wrote an article about the different kinds of feed readers and listing a bunch, maybe that's useful: https://lighthouseapp.io/blog/feed-reader-deep-dive

Same question, but for Android and desktop / laptop too. Never used RSS much before, hardly, in fact, I don't know why, even though I first knew about it many years ago, but after reading this thread, I want to.

Lire app on iOS does optional RSS caching of full text and images for offline reading.