They are a robust breed, and have few health problems. Puppies should be checked for deafness. Bull Terriers can have patella problems, and skin problems.
Selecting a Veterinarian
Make sure you find a vet that does not dislike Bull Terriers. There is much "Pit Bull" phobia in our society, and for your dogs sake you will need to find a vet that appreciates and respects Bull Terriers. A Bull Terrier's ears and/or tail should never be cropped. Any vet that suggests they should does not know the breed!
Bull Terriers are a large boned, muscular animal. Their daily feedings may require more protein than many other breeds. Consult the breeder you purchased your dog from, or a rescue representative (if you have a rescue dog) regarding the diet they feel is best for the particular animal.
Tattooing & Micro-chipping
Both tattooing and micro-chipping of a Bull Terrier is a good idea. This will aid in recover of an animal if it is ever lost or stolen.