Translation:My doctor wants me to neither smoke nor drink.
"Trinki" just means "to drink". Not quite sure what you mean by the addition of "regularly". "Drinki" means "to drink (alcohol) to excess". You wouldn't use it for sipping a glass of wine, but you might for finishing off the entire bottle and staggering down the road afterwards.
The doctor is saying what she wants you to do. She's saying "Nek fumu, nek drinku". Reported speech in Esperanto keeps the same verb ending as what was said originally, so for instance "He said he was going to Paris" is "Li diris, ke li iros al Parizo", because what he actually said was "Mi iros al Parizo". After words expressing a wish or an order you use the imperative in this kind of sentence.