Ah! Here I go.
After you finish the series (depending on the type of person you are, this concept seems either unimaginably far out, or depressingly soon), first read Dunk and Egg, the novellas set within about 150 years of the present day. They're much lighter than ASoIaF proper, but provide very interesting context for certain events that are otherwise in the background.
Next, check out the World of Ice and Fire background compendium. It helps so much with internalizing the universe. I'd recommend it even be read mid-series, because I think it adds to the enjoyment of book 3, which everyone famously hates, except I can't remember exactly what spoilers it contains. Nothing huge, because it's about the past, but you never know. It also has gorgeous pictures.
Then, look for the Princess and the Queen, which is a bit like a Silmarillion to Martin's trilogy, if that helps you at all. Dense-ish, historical, like the World of Ice and Fire entries but written by Martin himself, and more detailed.
Then, after you've finished with canon, read the fan-written theory essays from Tower of the Hand (actually they published and sell an ebook that's floating around somewhere), a forum which is something like 15 years old and has Martin's stamp of approval. (Not the theories! He rarely comments on anything substantive.) Then and only then will you realize that you actually only thought you read the series the first time; clearly everyone else read it better. So you'll launch a reread.
Optional: ping me in 1-6 months and ask for clarification about all this.