1. Be extreme cautious about the diseases that can be transmitted from dogs or cats to human beings.
  2. Examine the puppy /kitten or dog /cat carefully and neatly, preferably take it to your veterinarian to rule out any existing diseases. Few may be infectious, can affect your other pets at home, if any !  Scope of carrying contagious & zoonotic diseases like Rabies, Leptospirosis etc. should be carefully assessed and inspected. 
  3. Take suggestions from your vet regarding deworming of your adopted pets as many worms also can be transmitted from them to us particularly to children. 
  4. Know elaborately about the precautions that you have to take after you have taken/adopted that stray animal to your house for safe handling without making yourself prone to any kind of infections transmissible from animals to human beings. If a quarantine period is essential to maintain, you need to follow that. 
  5. Try utmost to avoid any kind of bite, scratch, lick on your injured/broken skin or on fresh cut of your skin unless this quarantine period is completed and you have completed its all possible immunization including anti rabies vaccine.
  6. An adopted stray dog or cat may have many complications and weakness regarding its health. Handle it delicately along with all possible personal health care measures as said above to treat it, so that it can survive and grow for a full health. 
  7. Complete its all possible vaccinations under indian standard in healthy clinical condition from your veterinarian.

I wish a better and healthy life for your adopted pet. Just follow these basics and go ahead. No adverse consequence will perturb you thereafter. Thank You.