You can't guarantee a puppy will be healthy unfortunately, but you can do your best to ensure you have the best chance possible of getting a healthy puppy.
Some people feel crossbreeds are more healthy in general, but sadly the people who deliberately breed them rarely give a thought to temprement or health and as has been said, if you go for a rescue crossbreed pup, you will probably not have any idea of the parent's health.
Personally, I feel getting a purebreed pup of a breed with few health problems, from a very good breeder gives you the best chance, but like I said, there is no guarantee.
For pedigree dogs, KC registration should be the minimum requirement. However, puppy farmers and irresponsible breeders still KC register, so you do need to be careful. Find a breed you like the sound of and then contact the breed club in your area to get details of breeders who have a litter planned. The best and most consciencious breeders will have a waiting list, so be prepared to wait for a pup, but they will be happy to meet you when you are ready, and in fact will probably insist on it before they agree to put you on their waiting list anyway. Going via the breed club means you are more likely to find a good breeder, but also mak esure they have done all the necessary health tests for their breed, things like hip and elbow scoring, any DNA or blood testing or eye tests required for that breed. Some things can be guarantee'd. L2-HGA in Staffs, for example, is a recessive condition which can be tested for and if both parents are clear, or one parent is clear, the puppy cannot have the condition. Other things like hip scoring do not guarantee the pup will not get hip displaycia, but as long as the parents have good results can mean it is far less likely, especially if grandparents also have low scores. A good breeder will probably have generations of parents, grandparents and grandparents of a puppy you are likely to get and can show you these and give you an idea of longevity of their lines. They should welcome any questions you may have and answer honestly and openly. They should also ask you quite a few questions.
Make sure that breeding dogs and bitches (parents of pups) are vaccinated and that both Mum and pups will be wormed, and ask what with. Find out what food puppy will be weaned on, it should be a quality brand, a good start is better than a bad one. Pups may also be frontlined, but not all breeders do this and tbh, there is no dogs the breeder has have fleas. The breeder may have had the first vaccination done, or may insist on keeping the pup until both vaccinations have been done. They also may not have had any vaccinations done. I do not have any pups vaccinated before they leave me as I (with advice from my previous and current vet) do not believe puppies should have their first vaccination before 9 weeks old and for socialisation reasons I believe they should leave me before that age if possible. Also, some vets use different brands of vaccine, and will insist on starting the course again if the breeder's vet uses a different vaccine. Plus, the puppy should still have some immunity from Mum at the age it leaves the breeder, and should not have been put at risk of any diseases which can be vaccinated against, and can be protected from the same diseases once with the new owner until they can be vaccinated.
Also, make sure the puppy has been properly checked over by the breeder's vet before you take it. Not all breeders do this, even otherwise very good ones, but I think if health is something you are especially worried about, it would be wise. Good breeders would take back, swap or refund a pup foun d ot be sick once it had left them, but that's obviously not the point.
Finally, do remember that things like cancer, epilepsy/siezures etc can occur even if the breeder has done everything they could to breed healthy puppies, but you can go a long way to prevent or help prevent a lot of potential problems.