We observe that the world (appears to) exist, and using what we observe, we can tell about the world's behavior. If the universe that we live in were just an illusion, then science would only tell us about the behavior of the illusion.

Einstein's theory of relativity holds that space and time are not separate things. From where you stand, there is a region of space and time expanding in all directions in front of you at the speed of light that is in the future. Similarly, there is a region of space and time expanding in all directions behind you at the speed of light that is in the past. We call these regions of space-time "light cones", because if you were to graph them in a four-dimensional hyperspace using time as a dimension, then they would be cone-like shapes in this hyperspace with you at the apex of each cone.

Events that take place outside of your light cones might be in the future, the present, or the past, depending on your state of motion. This is why time is relative in Einstein's theory, because events that are causally disconnected from you (you can't influence them and they can't influence you, due to the fact that information cannot travel faster than light) could be in the past, present, or future depending on how you move. Events that you can influence, however, are in the future, while events that could have influenced you are in the past.

Yes, Time exists. You can take a chicken and make chicken soup, but there's no way for the chicken soup to turn back into a chicken. That's an example of the 'arrow of time.'

Entropy requires time to have a direction, which seems like a pretty good indication that time exists:

"Entropy is the only quantity in the physical sciences that requires a particular direction for time, sometimes called an arrow of time. As one goes "forward" in time, the second law of thermodynamics says, the entropy of an isolated system can increase, but not decrease." From Wikipedia