Before people had any idea why anything happened, or how, they could at least find out what happened. People watched plants grow, they watched animals grow, they watched the sun go up and the stars and planets go around, and the seasons change, and they figured out a great many things about what happened. In fact, people of the Stone Age probably knew more about what happens in nature than you do, because that was more important for them than it is for you.
We often think today that people who lived a long time ago did not understand what makes plants grow, or where babies come from. But this is really not true. Anthropologists have never found any people on earth who didn't know both of those things very well. Anybody who looks around her will see that plants grow where seeds or spores fall on the ground, and that babies come from a mother and a father.
Constellation of Orion; thanks to Douglas Cooper
Another form of observation was the development of counting. As far back as the Paleolithic (Old Stone Age), about 35,000 years BC, people were carving marks on sticks to help them remember how many bags of fruit were in that basket or how many people were in the band. The Sumerians, about 4000 BC, were probably the first people to develop a way of writing numbers down more efficiently.
The Egyptians and the Babylonians were both very accurate and interested observers of natural phenomena. They identified and named the different constellations of stars. They also observed less obvious things, like the Pythagorean Theorem.
The Greek philosopher Aristotle brought the Greek interest in bringing order to the natural world to scientific observation, in the 300s BC. He tried to organize all of nature into a system that made sense, putting similar things together into categories. Aristotle got his friend Alexander the Great to send him back samples of new plants and animals that he found on his military campaigns, from as far away as India.
Not long after Aristotle came another Greek observer, a doctor named Hippocrates. Hippocrates made many observations about sick people, and he and his students wrote down what they observed, and their ideas about what it meant.
Under the Romans, Pliny the Elder around 70 AD collected masses of information into his Natural History, though mostly in order to show how big and powerful the Roman Empire was, that it had all these things in it. Pliny didn't really care what it meant.
A Roman doctor named Galen also added a great deal to Hippocrates' observations.
After the rise of Christianity, there was not much scientific observation going on in Europe for a while. In the Islamic Empire, from 700 AD on, on the other hand, people were very interested in scientific observation. Some people were interested in plants, and copied the Roman works while adding any new information they could find. Others, like Ibn Riza and Maimonides, were doctors, and added a lot of new observations (and also treatments) to those of Galen.