The current state of medicine in Star Trek is largely driven by the whims of the writers, but Brad Dehnert
gets most of it right. Here's a few clarifications.Surgery is very advanced, small issues are non-existant, but the remaining diseases are far more harmful and virulent.
Problems like lacerations and bruises are completely treatable in seconds. A tool called a "Dermal Regenerator" has become a staple of later series. Its widespread and multiple appearances place it as a common first aid device.
Jonathan Archer's father suffered from the fictional Clarke's disease. Supposedly this was cured by Dr. Noonian Soong after his death. His cure was saved, although the techniques detailed would require genetic resequencing (a banned practice among humans). This aversion to gene therapy has continued in humanity for nearly 150 years, and people show no sign of people rethinking their positions.
Early TNG was heavily influenced by Roddenberry. This spirit of optimism was reflected in the throwaway line by Beverly Crusher, headaches were a thing of the past, when the doctor was lecturing the captain about not coming to see her sooner. ("The Battle", TNG - Season 1, Episode 9)
As of the 24th century they can regrow limbs back. Rom's son Nog loses his leg in the DS9 Episode "The Seige of AR-558. It later became a small story arc which carried over a number of episodes. Since they had to return to a Starbase to have access to the right technology, it's probably something that isn't easily available.Pure conjecture here, but I'd assume high end Galaxy class (and later) vessels would have this. However, there have been a number of times where the Enterprise D didn't have adequate facilities to treat a medical emergency. One should note that there are no humans ever shown as amputees in Star Trek. So although it's probable people still lose limbs, treatment for it is widespread enough so anyone can eventually gain access.
What's notable is although there have been a number of genetic ailments there's no mention of cancer in the entire franchise run (admittedly, to the best of my recollection). In keeping with the Roddenberry theme I'd be shocked if he wanted it in his universe.
Medicine will seem to come a long way in the next 300 years. Really though, this answer's just a cheap excuse to show one of my favorite scenes featuring McCoy.
Also, there's pills that can repair damaged kidneys!