For the Maya, the Long Count calendar was in use in the BCs and continued up until about A.D.1000 when the last Long Count date was recorded at Chichen Itza. The Long Count records time from the date August 11, 3114 BC which was considered the beginning of the last era. Not clear on whether this is the beginning of the world, or just the end of the previous 13 bak'tun cycle, like what happened in 2012. Think of 13 bak'tun cycle as a millenium or similarly long period of time in our calendar, not as some world ending event. Some Maya texts refer to the recreation of the world at the 11th August date where the Long Count starts, but whether this is literal or a religious/ritual recreation is not clear. The city of Palenque in particular has a lot of references to this start date and ties the founding of the city back to it (not that the city was found then, but it compares the formal founding of the city to the ritual recreation of the universe at that start point).

Depends on the society. Some started counting from an arbitrary year, like we do with AD, some counted the years of a particular reign (Japan used both these systems for example), while the ancient Romans for example identified the year with the people who were Consuls at the time.

"Did people born in BC times who within their lifetimes switched over to AD times know that they were in AD?"

I'm not sure I understand your question. What do you mean by "know that they were in AD"?

Did you mean "did people realize Jesus was born", or "did people realize Jesus was born, and this was such a momentous occasion that eventually, their calendar would be changed", or even "did people born before 1 AD switch to AD reckoning within their lifetime"?

The first two questions conflate somewhat: first of all, you need to know that the person from whose birthdate the AD calendar would start counting even existed in the first place. While the general consensus is that Jesus most likely did exist (see the FAQ for discussion on that, for example: this question), it's very unlikely many people outside of the Roman province of Iudaea had ever heard of him during his lifetime. And even if you did, chances were that, to you, he'd just be yet another Jewish preacher of which there were many during that time.

But let's say we have someone born in 1 BC, and living a long life until 85 AD or so. In that case you might have heard the name, depending on where you lived. After all, the Christians were by that time a small religious group that had seen its share of persecutions. Even then, it's highly unlikely you'd consider the birth of Jesus important enough to start a new calendar, even if you were a Christian yourself.

The counting of years from the birth of Jesus wasn't really a thing, not even among Christians, until Beda Venerabilis introduced it in the 8th century (though he did not invent it himself; he built on earlier work by Dionysius Exiguus, who had lived in the 6th century).

The Greeks had a few systems. What was generally agreed upon was to mention how many years after the last Olympics had elapsed. So something like "3 years after the 27th Olympiad..." And so on. This is very handy for us because it allows us to peg dates down to the precise year, something unheard of in most fields of ancient history. Usually a decade is a pretty close guess in these circles.

Of course, Athens being Athens they had a different system. They would elect an official and then they'd say "in the 2nd year of so and so's administration..." Which is more annoying because we can only pin it down to two years due to confusion about some administrators service times.

But as commenters have said, it really varies from culture to culture.

No, that's not dumb. Despite not having a background outside your culture, you just realized something important implied by the whole BC/AD business.

To begin with, in the year 1 (there is no year 0 in BC/AD) no one knew anything was going on and paid no attention to events in Palestine in 95%+ of the world.

For starters, the whole Before Christ/Anno Domini business was not even invented until AD 525, by Dionysius Exiguus, based on a traditional idea of when the Christian savior was born, which is now considered more or less off. So no one born in BC times was still alive five centuries later when the system was invented. On top of that, it didn't even become popular until around 800. Naturally, no one used it who wasn't Christian.

Muslims count forward from the year of the Hijri, 622, so this is 1436–1437 to them, because they are using a calendar of 354 days -- year length doesn't necessarily match up between different old systems. Lunar calendars like the Muslim and Hebrew can be quite a bit off from solar calendars.

In the Hebrew calendar this is the year 5775–5776, dating from the beginning of the world. The Byzantine calendar used in the Eastern Roman Empire and in the Eastern Orthodox Church into the 1700s figured this differently, and say this is 7523–7524 since the Creation.

Buddhists figure from the time of Gautama's enlightenment:

In all Theravada traditions, the calendar's epochal year 0 date was the day in which the Buddha attained parinibbāna. However, not all traditions agree on when it actually took place. In Burmese Buddhist tradition, it was 13 May 544 BCE (Tuesday, Full moon of Kason 148 Anjanasakaraj).[3] But in Thailand, it was 11 March 545 BCE, the date which the current Thai lunisolar and solar calendars use as the epochal date. Yet, the Thai calendars for some reason have fixed the difference between their Buddhist Era (BE) numbering and the Christian/Common Era (CE) numbering at 543,[4] which points to an epochal year of 544 BCE, not 545 BCE. In Myanmar, the difference between BE and CE can be 543 or 544 for CE dates, and 544 or 543 for BCE dates, depending on the month of the Buddhist Era (as the Buddhist calendar straddles the Gregorian calendar—currently from April to April).

Is that confusing enough?

That makes this year about 2559 for them. But in the Thai solar calendar it's 2558.

To a Classical Roman, this would be ab urbe condita (from the founding of the city (of Rome)) 2768. The Assyrian calendar, still in use, starts from the founding of the temple in Assur, and this is 6765 to them.

In the Korean calendar, we are now in 4348. In two of the different Hindu calendars this is (Vikram Samvat) 2071–2072 or Kali Yuga 5116–5117.

On the other wing, in the Japanese calendar this is Heisei 27, the 27th year of the Heisei era. They start counting over again pretty frequently. This was the ordinary way to do things in many cultures. The earliest dates we have, in Mesopotamia, always are "In the xth year of the reign of King So-and-so." Getting those in order, and matching up reigns between different states is just one of the fun things in archaeology.

Then there's all the other methods mentioned in other answers, using archons or Olympiads or other markers.

So people "in BC" were already busy using their own calendar forms, and counting up years since the creation of the world or the founding of a city or temple, or since they last changed kings. There was no switchover, because no one knew there was anything happening. Centuries later, a scholar said, "We really should be taking note of this all-important event, not the years since the founding of Rome or even since the creation of the world," and came up with a back-dated date. It then took a few more centuries for the designation to start trending.

If I may build on this, what would ancient peoples use for smaller measurements of time? When did 60-second minutes/60-minute hours start to be widely used? How was the day divided or time referred to in say, classical Greece?

Thanks for all of the great answers guys! I think that /u/Zither13 understood what I was asking pretty clearly. As for all the rest, I still learned a good lot of how years were counted and how cultures' calendars worked. Thank you!