I've actually answered a question about time zones before, which you can read here, but the question about the "average person" is an interesting spin on that.

To quote from that other answer:

Hipparchus of Nicaea (c. ~190-120 BCE) seems to have been the first person to propose using a grid system to find the position of cities (and other places) on a globe, which implies an understanding of longitude. He built on earlier work by Eratosthenes of Cyrene (c. 276-194 BCE), who had mapped the known Earth, including finding its circumference. Hipparchus' method of finding the longitude of places was to use the differences in timing of lunar eclipses at different points on the globe to calculate the difference between local time of those points; the drawback is that there was no accurate-enough method of timekeeping to lead to useful calculations. Edit to clarify: The difference in local time between observed beginning and end of the eclipses would serve, essentially, as a way to understand the longitude between places.

The knowledge that local time would be different at different points on the globe is what led eventually to what's called "the discovery of the longitude" in around 1760 or so, when two methods of reliably finding longitude using time were discovered and implemented.

Now as far as the average person part of the answer, we know from books of sermons that medieval priests would be comparing the earth to a sphere fairly frequently -- so much so that it's quite reasonable to assume this is an analogy their parishioners would be comfortable with, meaning that it's likely ordinary people thought of the Earth as a sphere. Also, /u/sunagainstgold wrote a good answer to this recently, which I unfortunately am not finding, but I have written about it in the past as well. The tricky part is the knowledge of the earth rotating relative to the sun; which we know that scholars knew of, but I don't know of in relation to common people.

For example, were sailors or other people traveling long distances aware that they were traveling through "timezones"? I'm assuming probably not, as the change would be very gradual.

This is also in the linked thread, but at least in the English/British navy, the naval day started at (observed) high noon; the ship's watches would be reset then. (I mean watches in the sense of working times, not pocket watches or anything like that.) The watches would be timed with a half-hour hourglass, so the glass itself would be reset at local noon. It was quite possible to sail fast enough to gain or lose a few minutes a day, but it might be unusual -- for example, during their famous dash to the Caribbean in the run-up to Trafalgar, Villeneuve's fleet traversed about three of our modern time zones in about five weeks.