I don't have any knowledge about cats or other pets, but we brought our two 40+-pound dogs from the US with little problem.
In Valencia, it is common to see people walking dogs of medium to small size, and most families will have only one. It is rare that I see a dog off-leash unless they are exceptionally well-trained. There is an off-leash dog park not far from our flat, and occasionally we will see other people and their pets there, however, I've been there all hours of the day and it is rarely filled.
Recently (maybe the beginning of the year), the public bus system began allowing dogs but I've not seen any on the metro. The taxis have no specific regulations and it is up to the driver if they will allow dogs.
Dogs are often with their owners at outdoor seating at the cafes and smaller dogs--those that could lie on your lap and be hidden under the table--will sometimes be seated inside.
In department stores, it is acceptable to bring in small- to medium-sized dogs, but I've never seen any larger breeds.
Dogs are not seen in the grocery stores, but most have a place to tie your pet outside the store until you finish your shopping.
We could only find an unfurnished rental with our dogs, but I do have friends with 1 dog that have lived in a few furnished rentals. This is owner specific.
We found a vet within a 15-minute walk that is great and speaks English well. You can find basic food and treats at the corner markets, and the larger department stores and full-sized supermarkets have leashes, harnesses, toys, etc. Our dogs eat Beyond Salmon dog food and the 7.5kg bag (about 16lbs) is €30 from Carrefour. We get our pets' meds from the vet and also take them there for nail trims.
Some of the traditionally "more aggressive" breeds require a special license, insurance, mental health screening (for the owner), and muzzles. I don't have much information on that, but it is something to consider if you are getting a dog here. Your vet will be able to point you in the correct direction.
The mosquito breed that lives on the Mediterranean Sea side of Spain do not carry heartworm, so that is not a necessary medication (per my vet). However, there is a sand gnat that carries leishmaniosis (almost always fatal to dogs) so you need to keep a preventative, medicated collar on your dog; these have a 6-8 month use life.
