The difference between a clinic that already knows your animal and one meeting it during an emergency is considerable. Register in the first fortnight, while nothing is wrong.
What to do
Find a clinic within a reasonable drive, take the animal in for a routine check, and hand over the vaccination record so they hold the history. Ask what their out-of-hours arrangement is, and save the number where someone else in the house can find it.
Ask about exotics specifically
If you have a bird or a small animal, ask directly whether they treat that species. Many small-animal clinics do not see birds at all, and finding that out at eleven at night is the wrong time.
Keep the paperwork together
Booster schedules run from the original dates. If you move clinics, or move city, that history is the first thing a new vet will ask for.
What a first visit should cover
Weight recorded against a target, a general examination, the next booster date, and a conversation about diet. If diet does not come up, raise it - most first-year problems start there.