Haha, this is a question we've been pondering for some time in my family. My grandfather was an avid reader, and the gene has been passed through to me.
Honestly, I think you shouldn't start to think that there are some books your library 'must' have. English literature does have its classics and canon, but honestly I'd say the best way to build your own library is to buy whatever interests you the most at the time, read it, and keep it in your library when you're done. Don't try and stick to a schedule of any sort, it'll only bore you in the end.
Also, always note down the date of purchase of the book and your name, on the first blank page of the book itself, and somewhere else. Over time, you'll start to see patterns emerge in what you're reading and it'll help you make all sorts of connections. Today in 2015, I can pull out a musty old ledger and see what my grandfather was reading in the 60s. In its own way, it's a window into his mind.
I second foliosociety.com if you can afford it, but to be honest the vast majority of our library is plain old Penguin paperbacks. My granddad used to cover them with clear plastic film, it's something we continue to do. The covers are usually beautifully minimalist, and with a bit of care they can last a very, very long time.