Most health testing should be done on the breeding pair before they are even bred. Many of the typical tests you want to see can't even be done on puppies—the health of the breeding pair will tell you more about the possible health of the puppies they produce.
For a puppy, you should make sure he is up to date and on track with getting his puppy shots at the appropriate time. He should also be checked for worms. You need to take him to your own vet within 2-3 days of when you bring him home, rather than rely on the breeder's word about a vet check.
Be aware too, that the promise of a health warranty from the breeder isn't really worth very much. Even if your puppy turns out to have major problems say, 6 months after you get him, the warranty likely says only that you should return him for a trade—if the breeder is breeding substandard dogs, a trade-in is unlikely to be much better health-wise than the puppy you already have—and who wants to bring back a dog they have become attached to anyway, especially considering that a disreputable breeder is likely to just cut his costs and euthanize a puppy with severe health problems rather than seek treatment for him.
These are the tests you should look for—done on the parents:
1. Cardiomyopathy - a Holter monitor EKG and an ultrasound/sonogram done by a veterinary cardiologist within the last year before breeding
2. Von Willebrands via DNA test
3. X-rays for hip and elbow dysplasia, certified by OFA at or after the age of 2 years
4. Blood panels for thyroid by an OFA-certified laboratory. As well as a blood panel for the liver. Within the last year before breeding
5. Eye exam done by a veterinary ophthalmologist (CERF exam)
This should all be up to date and with acceptable results.
Testing is expensive, and are an area that many not-so-reputable breeders will skimp on. If you are not seeing the above tests with good results for both parents, find another breeder.