The answer to your question depends on which ancient civilizations you are studying.
Most ancient months began with the first siting of the new moon.
The Ancient Babylonian calendar had months of alternating 29-30 days with a "catch up" every 3 years.
The Mayans and Aztecs had a very elaborate solar calendar that told the months and different epochs of time.
The Egyptians told their months based on the moon and the annual flooding of the Nile river based on the location of the Dog Star, Sirius.
The early Europeans told time and seasons using an elaborate stonework clock on the order of Stonehenge.
The Romans actually told the time using a sun-dial during the day. The Chinese used a type of water-clock to tell the passage of time.
People in ancient civilization began to understand the concept of time by observing repetitive changes in the environ that took place with fair degree of regularity. First of such changes observed by them was the cycle of sight and darkness that occurred because of rising and setting of sun. This gave them the concept of day, which was the first unit of time they begin to recognize.
Next repetitive natural phenomenon that helped ancient people to understand and measure time was the changing shape of the moon. This followed a cycle of approximately 29 and 1/2 days.
Subsequently they started to recognize a year as a cycle of changing season. They were also able to link this yearly cycle with the changing position of the sun in the north-south direction.
With improved cooperation and technology, people felt the need to measure time more accurately in units smaller than hour. To be able to do this the, the ancient people first recognized the point of time when the sun is at its highest point in the sky during a day as exactly mid day, and a time exactly midway between two mid days as mid night.
Having thus defined a day exactly, they divided the day in smaller periods. For example, we speak generally of time like morning, afternoon, evening, and night. Some civilization developed more standardized system. For example, ancient Indian system divided the complete cycle of a day and night in eight pahars.
With advancing necessity and technology the time was divided in smaller units such as hours, minutes, and seconds. Although today we the system of hours, minutes and seconds throughout the world, in older times different civilization had different systems.
With division of day in smaller units came the need for measuring these time periods. The most reliable of these method was the sundial. As sundial uses movement of sun from sunrise to sunset as the basis of measurement of time, it gave most standardized time measurement. However sun dial worked only in daylight. Therefor other method were developed that relied on processes where the pace with which these processes occurred could be controlled with some degree of standardisation. This resulted in devices like burning of rope with knots, lighting of candles with graduations, dripping of water from one container to another, and hour glass.
In eighteenth century scientists observed that pendulums of a particular design have standard period of oscillation. This property was used to build clocks using movement of pendulum to measure time. These clocks were further improved and smaller watches were developed using movements controlled by action of springs rather than gravitational force.