Most city-centers will have a hospital. Ones out in the islands—South Pacific for instance—tend to be good and cheap, but in most cases you must be self-sufficient. That includes a well-stocked medical supply, and the know-how to fix yourself up.
We carry a general assortment of plasters/band-Aids, non-stick dressings, bandages, adhesive tape, non-latex gloves, antiseptic wipes, stainless steel scissors, safety pins, tweezers, ibuprofen, aspirin, antihistamines, seasickness meds, ear drops, eye wash, insect repellant, hand Sanitizer, staple gun (for wounds), topical anesthetic, zithromycin (bacterial infections), oxycodone/acetaminophen (severe, short term pain), sunscreen (SPF 30), aloe vera gel with lidocaine (for burns) etc.
If travelling into Malaria endemic areas, we carry rapid diagnostic tests. The test is 90% accurate and quickly diagnoses if Malaria is present, and if so, which one you have, so that treatment can commence quickly if needed. We only carried tests, because we were close to hospitals, but if traveling to islands with no facilities it is a good idea to also carry antimalarials.
In case of a serious injury, you must have the means to get yourself to the nearest clinic. We use DAN Boater, a repatriation insurance that covers helicopter/transportation fees.
We find that medical apps for mobiles, and FM army field manuals, are helpful when dealing with minor burns, cuts and various aches.