Hey Rowan,
Here's how I would do it:
First, you'll need some kind of identifier for the button in your HTML code. It could look like this: <button class="zoomButton" type="button">Click me!</button>
Now you need to attach an event in your Javascript code (in this case it's a click event). I usually use jQuery for this. Here's an example;
jQuery(".zoomButton").click(function(){
//this is where you put the leaflet functions, like map.getZoom(), map.getCenter() etc...
});
You'll want to save your zoom/center to variables, and push it wherever you need it. It's probably super bad forum form for me to provide you with the working code, but I remember how frustrated I was when I was trying to get help learning code. I highly recommend learning the vocabulary for what you want to do so that you can find the tools yourself, and any beginner's Javascript or jQuery reference can help you do that. I started by using the FREE courses on
codeacademy.com. I also thinks it's really helpful to test your functions/events using alerts first to make sure they're actually triggering, which makes debugging easier early-on. You'll want to check the type of the return value, too, to make sure you're getting what you want out of your efforts (alert(typeof [insert your variable name here]); would be a good start).
Anyway, I hope that was helpful and not pedantic. Good luck!