Whippets are one of the healthiest breeds. They require the same routine care as any dog: trimming toenails, cleaning of ears and teeth, occasional baths. They have to be kept up to date on shots, free of heartworms and fleas, and checked for intestinal parasites regularly.
Though all breeds have some genetic defects, none have been proven to be a serious problem in whippets thus far. Some isolated incidents of deafness and some genetic eye defects have been reported, but they are rare. There have been a few reports of a bleeding disorder called von Willebrands. Many breeders do test for those and screen for eye and hearing disorders.
The most serious health question facing us now is cardiac. The Whippet Health Foundation is funding a long-range study to determine whether a genetic heart problem is common in the breed.
Sighthounds are sensitive to anesthesia and other medications. Partly because of their low percentage of body fat, these dogs are extremely sensitive to some very common drugs; what may seem like a normal dose for a dog of his weight could easily kill a sighthound. This is certainly not to say that whippets cannot be safely anesthetized or that they should not take prescribed medicines; just be sure that your vet is aware of sighthounds' special requirements and that he knows which anesthetics are safe.
Finally, Whippet skin is thin, rather tightly stretched, and poorly protected by the scant coat. What would be a small cut on another breed can become an ugly tear on your whip. Unless blood is actually spurting out this is not an emergency but may require a vet to stitch him up so the injury will heal.