As a commenter stated, prevention is the best way to protect your current livestock. There are some illnesses that are easy to treat such as ich and some that can wipe out an entire tank. I'll use ich as an example because it's by far the most common illness freshwater aquariums will encounter.
The best defence is using a cycled quarantine tank with all new arrivals and do not introduce them into the display tank until they have been quarantined at least a month, ideally 6 weeks. This time-frame is important because fish can carry certain pathogens such as ich, but not show any symptoms for days to weeks. If they do need treatment for an illness, the month to 6 week time-frame applies from when the treatment period ends. If you don't have a pre-cycled filter or tank, then quarantine usually isn't an option as you will end up cycling the tank with the fish in it, and the ammonia and nitrite created during the cycle will harm or kill most tropical fish.
If pre-quarantining isn't an option, always make sure to buy the best stock possible. Make sure that no fish in the tank has visible signs of illness. If the store has all the tanks plumbed together, this would technically include all of the tanks using the same water. Try not to purchase fish close to when the store receives shipments as the fish store is going to be dealing with the same possibility of sick fish with no symptoms. The stress of shipping increases the chances for opportunistic infections as well. Obviously this still isn't full proof and is still more or less rolling the dice.
Once you have a sick fish in the tank it gets more complicated, especially if you don't have a cycled tank that can be used for treating the sick fish. First step is to try and identify the illness. For many illnesses, you can assume that if 1 fish has it, the rest have been exposed to it. Identification is important though because the treatment options are often very different and shotgun medicating is normally ineffective and can often do more harm than good to the livestock.
If you are dealing with ich or another pathogen that is going to infect everything in the tank, you will want to begin treating the whole tank. The treatment depends on what is infecting the tank and could be anything from simply elevating the temperature, or adding antibiotics, copper, or another aggressive chemical.
If you do have another cycled tank, and the illness isn't something that is going to affect the entire tank, such as a fungus or bacterial infection, or you have no idea what it is, it's best to remove the fish and observe and treat separately. If it ends up being something that is going to affect the whole display tank, return the fish and treat the whole tank. As far as whether it's going to stress them out to move, it definitely is, but for many illnesses the chance of survival without treatment is zero, so moving them is necessary.
If you don't have a cycled tank to use as a hospital the options are limited. You can treat the whole tank if you know what the fish has and the treatment isn't going to harm the other fish. If unknown you can take some basic steps to help the fish recover. Add kosher or other non-iodized salt at a rate of 1 Tbsp per 5 gallons, or 1 Tbsp per 10 gallons if there are live plants in the tank. If it looks like an internal infection use plain Epsom salt at the same rate instead of normal salt (Don't use both). Increase the temperature to the mid 80's assuming all the fish are tropical (not for goldfish) and add additional aeration if the fish begin gasping at the surface. Feed a good quality food, frozen is usually the best, with the addition of garlic. Beyond that without knowing exactly what to treat for, it's wait and hope.