The general rule for customs in the EU is that you're only customs processed at your final destination -- except if you leave the airport terminal at an earlier transfer point.
Usually customs will be located right after the baggage reclaim at the destination airport, and there will be signs directing you to choose between green lane (nothing to declare), red lane (something to declare), and blue lane (nothing to declare because your entire journey was within the EU).
(If you leave the terminal in Spain and have baggage checked through to France, you'll have to behave as if arriving internationally at both airports. Hold on to whatever documentation you get from Spanish customs, so you can convince French customs that the animals and your cabin luggage was already in the EU!)
I'm assuming that you have researched what you're doing properly and that your only worry really is to get in touch with customs at the right time and place. If not, beware that traveling across borders with live animals can involve significantly more involved and restrictive rules than moving either persons or goods across the same borders, for veterinary safety reasons. Even when transporting animals between different EU/Schengen states, there are restrictions that must be observed.
In particular, just because your destination airport has customs processing doesn't necessarily mean it's equipped to do veterinary border checks (which you may or may not need based on the species of the animals and other factors).
Be sure to do your homework in advance.
For completeness, immigration checks are different from customs and happens at your first landing in the Schengen area (with variations possible if your itinerary involves both Schengen and non-Schengen Europe).