The Earth rotates around its own axis once every twenty-four hours. The Moon, on the other hand, rotates once around its own axis every 28 days, and once around the Earth in that same 28 days. The end result of this combination is that the same side of the Moon is always facing the Earth. As the Moon moves to be directly above a different portion of the earth, its face also turns at exactly the same rate, so that only one hemisphere of the Moon is ever visible from our home here.
If the Moon turned at any other rate (either faster or slower), we would eventually see all sides of the Moon, and what is currently the lunar far side would be a much more familiar sight to us. If we spun up the Moon to one rotation every 24 hours, how dramatic would this be?
I’m assuming that we’re not changing the Moon’s orbit here - so that the Moon would still orbit the Earth once every 28 days. This means that the rising and setting of the moon would happen in the same way as they do now - slightly later every day, and the phases of the moon would remain the same, because the phases are simply the combination of the Moon’s location in its orbit around the Earth, and what fraction of the near side of the Moon is illuminated by the Sun. So we would still have a new moon and a full moon about once per month. What would certainly change is which portion of the moon is illuminated.
Speeding up the Moon’s rotation so that it spins once every 24 hours is a pretty dramatic change. That means the Moon has to rotate the full 360 degrees of a circle in 24 hours, which puts us at 15 degrees of rotation every hour. That may not sound like a lot, but over the course of an evening, which we’ll say is an average of 12 hours (half of our Earth’s 24), that means that the Moon has rotated by 180 degrees. A full moon could rise with the familiar near side facing us, and by the time it sets, 12 hours later, we’d be looking at the unfamiliar jagged territory of the lunar highlands - what is currently the lunar far side. In a six hour period, you’d expect the Moon to rotate by 90 degrees. If you were in a half-moon phase, where only half of the Moon’s face is illuminated, you would expect that illuminated portion to change completely, twice over, every time the Moon rose above the horizon.
However, if the Moon truly did rotate once every 24 hours, the two sides would probably look much more similar to each other than they do now. Part of the intense cratering of the far side of the Moon is because it is constantly facing “outwards” towards space, and it’s an easier target for interplanetary fragments of rock to hit, than the somewhat protected, Earth-facing side. If the Moon rotated faster, these meteoroids would have a pretty even chance of hitting any face of the moon, and the cratering would probably be more evenly distributed.
It’s fun to think about how this kind of situation might have influenced our calendars - since our months are roughly based on the lunar cycle, perhaps we would have used the appearance or disappearance of certain features of the moon as a smaller unit of time. But we certainly wouldn’t have grown attached to one side of the Moon - what we see now as the near side would be just as normal to us as the far side.