
You need to try and get Total T4, free T4, total T3, free T3 and TgAA antibodies as a minimum. Anything less is not sufficient for diagnosis. A lot of vets here can't get everything though, at the least try to get the free values of each in preference to the total, as this is what's actually available for the body to use. It's important to get the T3 as if there is any conversion problem (T4 gets converted to T3 so the body can use it), a T4 only or T4/TSH or TSH only test will not pick it up. It's rare in dogs, but it does happen - one of mine has this problem.
TSH is next to useless in dogs - the vet will doubtless try to persuade you otherwise, but don't be fooled. There's no exact figure, but roughly 20-40% of hypothyroid dogs have a TSH within range; Remy (my conversion issue dog) has always, always had a normal TSH. It took me three years to get him diagnosed because the vets don't know how to do it properly, and it was me who finally cracked it.
Ideally, you want to send samples off to the US for testing - Hemopet do a full panel, Michigan State Uni do a more thorough full panel, but it takes a little organising and a lot of vets aren't keen on doing it for some reason. Mine have never argued about it but a friend of mine has spent years trying to get them to do it and all the vets he's seen have refused!
If you want some vet-written information have a look on itsfortheanimals.com, look for the hemopet/Dr Dodds info - there are articles there on diagnosis, treatment and symptoms. Googling Dr Jean Dodds will find some info too, she is the foremost authority on thyroid problems in dogs. In each of my four hypo dogs, it's been her that has helped me persuade the vet at the time to trial them on medication.
On that note - when you get the results, get the numbers, the reference ranges and the units - Dr D can do interpretation for you. Do NOT accept 'normal' as a result, this is a misnomer - it simply means that the result is within the lab's range. All four of my dogs were within range at first testing, which is why Remy took so long to get diagnosed - the other three were almost straight away but by then, I knew what I was looking at and was able to push for treatment.