So, 'A New Dawn' is actually the first book in the Dawn of the Clans arc, which is the sixth overall series in the massive 'Warriors' universe. I see a lot of new readers get confused and think it's a modern sequel, but it's the opposite—it's a full-on prequel set way, way before the original series with Thunder, Shadow, River, and Wind Clans as we know them. It's about the very beginning, showing how the cats first came to the forest and formed the Clans from scattered groups.
It connects by establishing all the original lore and rules. You see why the Warrior Code exists, where the concept of StarClan and the Moonstone came from, and the deep-seated reasons for the animosity between, say, ThunderClan and ShadowClan. Characters like Gray Wing and Clear Sky are basically the founding fathers. Reading it gives incredible context to stuff that gets referenced as ancient history in Firestar's time, like the Great Journey or the reasons behind certain territory disputes. The connection isn't through continuing a story, but by building the entire foundation that the original series stands on.
If you've read the main series first, going back to 'A New Dawn' feels like uncovering the answers to mysteries you didn't even know were mysteries. Why do cats go to the Moonstone? Where did the concept of a medicine cat originate? It's all there. It makes re-reading the original books richer, because you understand the weight of the traditions the modern cats are upholding.