I've read the entire series. It's well worth it, but the hidden meanings in the story are unbelievably complex. Gene Wolfe simply doesn't like to come out and hold your hand and tell you the importance of what's just happened. He wants you to figure out some of the connections yourself. For that reason, readers have produced an enormous quantity of interpretive material. See, in particular, the thorough and searchable
www.urth.net. This is the most detailed resource, but it's not well-organized. I got hold of the very helpful Lexicon Urthus book, but that was only after I finished the series. That book is great for vocabulary and it also has a summary of all the major events in the New Sun series. The author has recently come out with a similar book for the third part of the series (the Short Sun), which is good, because that might be the most mysterious part--in good and perhaps also in some frustrating ways. Urth of the New Sun is pretty tricky as well.
This series is well-known for making the extraordinary seem commonplace. The writing is indeed exceptional, but I think the series isn't as well-read as it deserves to be, because it's full of intricate puzzles. Reading the New Sun books made me want to write my fiction in the first person. Mind you, Wolfe is also well-known for using untrustworthy narrators to add yet another layer of mystery. Is Severian lying at some points? There's lots of discussion of that as well.
If you like this genre, I'd recommend Jack Vance's Tales of the Dying Earth. That's a more amusing series. Some of the stories in Songs of the Dying Earth are also great.