Different cats will respond differently to the same infection, so you can't really go by the exact same symptoms in every case. It usually affects the gut but it can be widespread thoughout the body or the cat may have no symptoms at all. Cats are exposed to it from ingesting contaminated feces, infected prey or contaminated food. The incubation period is usually 3-5 days and bacteria can be shed for 6 weeks, even in asymptomatic cats. It can exist in the environment, and is resistant to disinfectants. You should clean the kitten's environment with a 1:10 bleach solution. Signs are vomiting and diarrhea, fever, abdominal pain, hyperactive bowel sounds (growly tummy), weight loss and no appetite. The tests to diagnose it are cultures of the stool and or blood. The prognosis is actually pretty good, with recovery usually occurring in 1-4 weeks, unless the cat has the infection in their blood. The antibiotics usually used are Baytril, Zeniquin or Septra (Bactrim). He will need treatment for 7-10 days and needs to take all of his antibiotic. You need to be aware that if he has FELV or FIV, he will be much sicker and less likely to survive.
So, you need to give him his medicine as ordered, make sure he eats and drinks and keep him warm and calm. You need to isolate him from any other cats and clean everywhere he has been as well as you can with a 1:10 bleach solution. That isn't practical for carpets and couches, but vacuum them well and if possible put anything you can out in the sun for a day or so. Clean his litter box daily with a bleach solution and consider throwing it away when he has recovered.
I hope this information helps and feel free to ask more questions. Becky