how to teach a puppy not to take things out of bins?

How old is your puppy? If she is like 8 weeks or something, they are going to be too young to understand. If she's a few months old or older, what I would recommend is that you purchase a squirt gun from Walmart and everytime she goes into your garden, say "AH", and squirt her. Don't squirt in the eyes or ears because that could be very dangerous. Squirt on the body. I have done that with my German Shepherd and trust me it works. Sometimes doing "AH" alone doesn't help so the squirt gun is absolutely necessary. Another thing you can do is take her water bowl and wiggle the bowl over her and let some water fall out on her. Actually - that is very effective. When my dog is misbehaving, sometimes all I gotta do is pick that bowl up and show him and then order him to come inside or get away from something and he will comply.

EDIT: in your case, you might want to use a squirt gun because your dog could associate her water bowl as a negative thing and might not drink from it. I have trained my dog to understand when the bowl is a good thing or a bad thing. He totally understands but that's a lot of extra training so a squirt gun is better.

Move the bins!!! Most puppies, especially Labs, will get into anything there is to get into. Apart from the mess, she could well get into something dangerous to her, like cooked bones and end up on the vet's operating table. When you have a puppy, the ONLY solution is to dog-proof your home and don't leave her alone and unsupervised to get into stuff like this.