As mentioned by @user282192, The second sentence should read "The doctors are trying their best to save his life." That is because you are talking about a specific group of doctors, and they are the subject.
"The doctors" is a noun group. The article "the" is a quantifier and it adds specificity to the noun, "doctors," so it's clear that you are talking about the specific group of doctors trying to save your grandfather's life, and not just doctors in general.
Since I'm assuming that the doctors trying to save your grandfather's life are the same ones advising him/you about his recovery, you don't need to completely re-introduce the subject in the third sentence. Here, it would be perfectly correct to use "They" instead of "The doctors."
Because the 2nd and 3rd sentences are constructed in a similar (parallel) way, it's still clear who the subject is, even if you use a general pronoun. It wouldn't technically be grammatically incorrect to use "The doctors" here again, but it sounds repetitive, and if I were proofreading this passage, I would suggest using "They" over "The doctors" to avoid repetition.