So just how old does a coin have to be, to be classified as ancient?
Generally speaking, modern world coins begin in the year 1600 (an arbitrary Krause Publication thing I think but I don't know for sure).
Coins made between 1492 to 1599 are generally considered Renaissance / Reformation era which was really the beginning of the modern era.
So I'd say anything made prior to 1492 is ancient broadly speaking.
Byzantine & Medieval coins are generally treated as a later era subset of ancient coins.
The oldest ancient coins are ones generally struck out of oblong lumps of naturally occurring electrum from parts of Greece that date back to the 7th century B.C.
The most widely collected are probably Late Roman Bronzes in uncleaned condition. You buy them in large lots on eBay and clean them yourself. It's a very cheap way to start collecting ancients. You can buy them for less than $1 each but you get what you pay for.
Collecting Roman silver & billon (debased silver) denarii and antoninianii of common emperors like Hadrian, Septimius Severus, Gordianus III & Philip the Arab is an inexpensive place to start for collecting ancient silver.
A common misconception is that any coin pre-dating 1900 is ancient. I often get unsolicited emails from people with coins they inherited from a grandparent like Morgan silver dollars from the 1800s who want to know what their "ancient coin is worth" and I tell them I don't deal much in 'modern' U.S. coins and you can just see their head spinning in disbelief.
Hope that answers your question.