Fatou, with a dog and two cats I've done this search twice now. For rentals, what worked was contacting landlords directly with a short "pet CV": the animals' ages, vaccination records, a reference from our previous landlord saying there was no damage, and an offer of a slightly higher deposit. Half the "no pets" listings said yes once they saw it. For the vet, I asked in the local dog-walking group in the park, where every owner has an opinion and the same two names kept coming up.
Two worries once we land: finding a good vet, and the fact that half the rentals seem to say "no pets". How did you sort both without too much stress?