It’s really easy.
You can download books in your kindle account onto your new kindle plus other devices, such as your phone or iPad etc. by using the using the kindle app.
In fact my daughter and I both have kindles, and Mr Moongirl too, all linked via my amazon account so we can share all the books between us. Well, in reality, my daughter and I share our fiction and some non-fiction and Mr M mainly shares some of the non fiction books (as he doesn’t share our taste in fiction, nor has an interest in cookbooks, etc.). The only downside, which is not a problem at all, is that I have to order all the books on my amazon account, but this is outweighed by the benefit of sharing them between us all.
It’s also really useful, say for guidebooks, to download onto a phone or tablet and screenshot maps and information pages which you can read on location without using WiFi or carrying around a heavy guidebook.
Also, non fiction books when read using the kindle app on an iPad, laptop, etc. appear in full colour and much larger too, so are easier to navigate and read, etc.