Working with larger animals. Watch
If you mean domesticated animals, then work experience with a vet will be your best bet, that may get you cattle, sheep, horses and pigs. In pure voluntary work, then you are probably looking at horse and donkey sanctuaries, no-one rescues animals we eat (sheep, pigs, cows). I suppose a city farm might have a few goats or sheep?
Equine is fairly easy to get if you go to a riding school. They will be able to teach you the basics, like tacking up, grooming, mucking out, feeding, cleaning tack etc etc. They're especially grateful for help over summer, which is also when it's busiest. If you're already quite horsey then you could go to a racing/polo yard, stud farm, or police stables. I would suggest phone or email to arrange these.
Farming can also be fairly easy to arrange. Lambing is great fun and I would totally recommend it, a lot of places will be able to offer you accommodation as well. The NSA publish a list of placements every year: http://www.nationalsheep.org.uk/next...ng-experience/. There are some other directories of where to get different farm animal experience here. For commercial/food producing farms I would suggest phoning, emails/snail mail tends not to get a response. Urban farms might be easier to get into if you are city-based, and although you may not be able to see the proper economics of running a proper farm, you'll probably be able to get experience with a wider variety of animals.
As for other placements, you could consider sanctuaries, aquariums and zoos. I disagree with the poster saying that you won't be able to get hands on, my zoo placement was very practical and I was able to see vets giving vaccinations etc etc. Zoos can get booked up very far in advance so I would suggest getting in touch ASAP. The BIAZA list of zoos that offer work experience is on the thread I linked to above.