Each to their own etc, but my choice is usually something like WWF for the reasons @steaksauce mentions. But, I would say that, as well as tackling the bigger picture, (the environment), there are already metaphorical fires which need to be put out. Famine, war, water shortages, homelessness, refugee crisis; all of which can be linked to climate change / the environment.
So, I think we need both. I thought the general consensus was that preventing climate change actually isn’t an option any more, so tackling the problems it is already causing is part of the work which has to be done imo. That includes things like the refugee crisis, water shortages, famine etc.
As for animal shelters, which don’t euthanise... I wouldn’t send them any money personally, no.
Pets, which can’t be rehomed, may be better off being put down in some cases. Much as I like animals, (former long term veggie and short term vegan), I personally don’t think keeping homeless pets alive, should be top priority. It sounds cold, but if one has to prioritise.
Wild animals are different though. Conserving wildlife is something I think is very important. Not because I like the cute, fluffy animals, but because they’re part of the ecosystem we humans are currently messing with, but upon which we rely heavily.