The ancient peoples in the near east, particularly the Babylonians and the Egyptians were the first known peoples to record celestial events in a systematic way. There is some evidence for earlier astronomical observations, but no proof of a systematic approach.
There are many phenomena in the night sky which repeat regularly. The Moon goes through its phases every 28 days or so. The seasons change the weather regularly every year, and the stars change together with them.
The Babylonians are responsible for many astronomical things that are still in use. The starsigns familiar from the superstitious field of astrology are originally Babylonian inventions. Our habit of dividing the hour into 60 minutes and the minute into 60 seconds is also ultimately from the Babylonians, who used a number system based on 60 (in the same way as ours is based on 10).
The earliest calendars that we know of (Babylonian, Egyptian) were based on the phases of the Moon and did not run in sync with the seasons. Some of these calendars are still in use with small modifications (the Islamic and Jewish calendars).
Later calendars were based on the cycle of the Sun, which means that they are in sync with the seasons. Our own Gregorian calendar is only a small modification on the Julian calendar, which was established by Julius Caesar around 45 BC, so even our calendar system has been in continuous use (with the small modifications) for over 2000 years now.
The Egyptians divided daytime from sunrise to sunset into 12 "hours". The length of these hours would of course change with the seasons as days grew longer and shorter. They did not bother dividing the night since the main type of clock was the sundial. Later as clocks developed, the night time was divided into twelve as well, and the lengths of these were fixed so that a whole day has 24 equally long hours. So also our clock can be traced back to ancient Egypt.
See wikipedia: Lunar calendar for a more detailed and relatively clear description of lunar calendars.
When one begins observing the sky, the first thing one notices is that the Sun moves across the sky as the day passes. This is due to the rotation of the Earth around its own spin axis.
After observing the Sun for a year, one can figure out that the Sun also makes a year-long circle around the background of the stars. This is due to the Earth's orbit around the Sun, which takes one year.
During one year, the seasons change on Earth. In the northern hemisphere, the days grow shorter in the winter and longer in the summer (this really is almost a definition of "winter" and "summer"). On the southern hemisphere, the same thing happens in the opposite sense. Southern winter coincides with northern summer.
The seasons are caused by the tilt of the Earths spin axis, relative to its orbital plane around the Sun. Due to this tilt, the northern hemisphere is tilted towards the sun in one part of the orbit, and away from it in the opposite part. These correspond to summer (towards the sun) and winter (away from the sun).
In between winter and summer, the axis is tilted sideways compared to the Sun. These parts of the orbit correspond to autumn and spring, when the days and nights are equally long.
The Moon orbits the Earth in a more or less circular orbit. The distance between the Earth and the Moon is very small compared to the distance between the Earth and the Sun. Thus the Earth-Moon system orbits the Sun practically as one object.
The Moon also spins around its own axis. This spin is synchronized by tidal forces so that the Moon always points the same side towards the Earth.
As the Moon orbits the Earth, it is illuminated by the Sun from various directions. This causes the visible phases of the Moon, making it shrink from a full bright disk into a narrow sickle, disappear completely and then grow back as a sickle in the opposite direction.
The Sun and the Moon happen, by sheer coincidence, to be the same size as seen from Earth. Thus whenever the Moon moves in front of the Sun, it covers it completely, causing a solar eclipse. One might expect this to happen every month, as the Moon orbits the Earth, but the orbit of the Moon is also slightly tilted, making the Moon mostly pass either over or under the Sun in the sky. Every now and then they are aligned and an eclipse happens. Predicting solar eclipses exactly requires regular observations of the Sun and the Moon for a fairly long time.
For a detailed description of the lunar phases, wikipedia: Lunar phase is a relatively good source (it does not seem to contain any gross inaccuracies at least).
See this picture for an illustration of the Earth and the Moon, to actual scale both in their sizes and the distance between them.