Well as early as 460 BC it was known that fevers had many different causes and were symptoms as opposed to actual diseases, and Hippocrates identified malaria as one cause. Egyptian (as early as the 4th millennium BC) and Greek physicians identified various effective means of alleviating fevers and headaches such as castoreum, and willow plant extracts containing salicylic acid and aconitine derived from wolf's bane (also known to be toxic in large doses like many medicines are) and much of that medical knowledge was adopted by Roman physicians. Garlic, zinc and other ingredients such as salts and extracts such as witch hazel known to be helpful today were known then and ancient physicians did understand that the best way to handle disease was through prevention (eating healthy, getting exercise, fresh air, all that good stuff).

However, ancient medicine also relied heavily on mysticism and folk cures including chanting, prayers, amulets and various ingredients (such as dung and animal ingredients) now known to be placebos or even harmful. Hippocrates believed that health depended on a balance of the Four Humours, Black Bile, Yellow Bile, Phlegm, and Blood, (a belief with roots in Egypt and Mesopotamia) and Humourism was adopted by the Greek, Roman and Indian physicians until the 19th century. Egyptian physicians often attributed ailments to demons and prescribed theophoric (religious symbolism) phylactic (containing written spells) and homeopoetic (animal parts that would lend the positive attributes of the animal) amulets and protective spells. Romans also were quite fond of prescribing amulets to prevent and cure illness.

A lot of medicinal research had to be observational so many false assumptions were made based on correllations and confirmation bias, but these were still intelligent people trying to figure out the most effective way to get people healthy using common sense and the tools they found around them.

Sources:

De Materia Medica by Pedanius Dioscorides

Natural history by Pliny

Hippocratic Corpus

On the Nature Of Man by Hippocrates

Edwin Smith Papyrus

Before people understood what bacteria and viruses were, they actually were pretty good at intuitively understanding that illness was usually picked up from the environment, and not randomly generated in the body. "Miasma theory" (bad air/pollution in Ancient Greek) dominated before germ theory. The general idea of miasma was that sickness hung around in the air, whether caused by death/decay, certain animals or plants, sewage, etc. You'll notice this isn't too far off from the truth.

This paper on the history of antipyretic (anti-fever) therapy answers your questions pretty well. Though it's from a medical journal, it's short and mostly written without medical jargon. Most relevant parts:

When Alexander the Great was stricken with the mysterious febrile illness that would take his life in 323 BC, the Babylonian physicians who cared for him prescribed cool baths as treatment for his unremitting fever

Pharmacologic treatment of fever also extends deep into the human historical record. Stone tablets dating to the Sumerian period describe the use of willow leaves by Assyrian physicians to treat inflammatory rheumatic diseases [3]. The Ebers papyrus, which has been dated to circa 1500 BC, verifies that the ancient Egyptians were also aware of the antipyretic property of willow leaves and used them to treat various inflammatory disorders. According to the papyrus, when a “wound is inflamed…[there is] a concentration of heat; the lips of that wound are reddened and that man is hot in consequence…then you must make cooling substances for him to draw the heat out…leaves of the willow.”

It's surprising, but ancient peoples had a pretty firm grasp on inflammation. Funny that you brought up treating a wound also, because that was largely the basis for early understanding of fever. The skin around a wound gets hot because it is infected; it stands to reason that the whole body feels hot when we're infected inside. Alexander the Great was suddenly stricken by fever (possibly due to malaria or typhoid), and even in those days, it was already quickly understood what it meant and how to treat it.