Bringing your pet or pets to another country can be a frustrating experience if you don't know what you need to do and how you need to do it. Even if you are purchasing a Pet here on the Orihuela Costa or the Costa Blanca or anywhere else in Spain for that matter, it is always best to seek adcive from those who know......i.e. a Vet or your local Town Hall.
In Spain, there are certain rules for bringing pets into, haveing them here and also maving them around Europe (in particular to the U.K. & Ireland).
Obviously many Countries have their own indiginous, bugs, pests and diseases, partibularly in a hot climate like Southern Spain, so you should get your pet to your chosen vet for a full healthcheck and to gain some valuable advice about what the partcular needs of your pet may be, from a possible change of diet, to some precautionary vaccinations to advice about paritcular avoidances when out with your pet (for example - the Processionary Caterpillar).

Vetting The Vets.
Chosing your Vet is akin to choosing your own Doctor or Dentist. You want someone who is frendly, professional and (hopefully) not too expensive, along with being easy to reach, especially in an emergency (there are special emergency Veterinary services on the Costa Blanca).

Here Is The Good News !
Here on the Orihuela Costa and Spain, petcare is well priced, Vets are friendly, most have excellent reputations, are highly professional and many of them will speak English, if you cannot speak Spanish. (but please try).
There are large Petcare emporiums (like Petsworld), who have everything that you will need to care for your pet, from Dog & Cat food & treats, fish of all sorts, beds, baskets, leads etc etc, all the way to in-house surgery and are very affordable and are highly professional.