Frogs are amphibians that do well in captivity as long as optimum conditions are met for them to thrive.
General - You can start by catching or buying inexpensive tadpoles and enjoy raising them to froghood! Or there are more exotic frogs available in the pet trade, from tiny colorful nonpoisonous dart frogs to African clawed underwater frogs to the large pac-man behemoths! Regardless of where your fancy lies, it is all about keeping the water uncontaminated and pathogen-free. Minimize handling them as their skin is quite delicate and if damaged, will allow a bacterial infection to take hold!
Veterinary Care - Sick frogs should be transported in a moist container and some of their tank water should be provided to be tested for parasites, bacteria and chemical contaminants. X-rays, bloodwork, and surgery can be used to diagnose and treat your frog to restore them to good health.
Remember to always wash your hands after handling any animal, especially reptiles and amphibians. See information from the CDC about reptiles and contagious salmonella infections.