This is a very interesting and unfortunately, a very difficult question to answer.
Unlike other numismatic areas, ancient numismatics has a diverse range of coins from a diverse range of cultures spanning over a thousand years.
This figure is unfathomable when you compare it to a "modern" collecting interest such as Australian pre-decimal coins which only span a little over fifty years.
This guide is not intended to provide a comprehensive answer to the question posed, but outline a quick sketch of the options available to you as well as their respective advantages and disadvantages.
One of the first things you must ask yourself before making any purchases, is what area of ancient coins interests you? As a new collector, you are prone to giving into temptation and purchasing every "pretty" coin that passes your way. This is a quick way to run out of much-needed funds and only leads to regret later on down the track.
Regarding ancient numismatics, the first question I would recommend asking yourself is what empire you'd like to collect from. There are several advantages and disadvantages of each but it all depends on your individual taste. I will run through these advantages and adisadvantages individually.
There are many advantages in collecting Roman coins. First of all they are cheaper. Don't misinterpret this statement though, there are many Roman coins that will be out of your price range, but as a general assumption this holds true. Compared to Greek coins, uncleaned Roman coins are always cheaper, third to 5th century coins are always cheaper than the "typical" greek coin, and even silver roman coins are cheaper than greek silver coins.
Also, Roman coins are easier to attribute and to follow generally. Roman Imperial coins have the emperor on the obverse 99% of the time and thus you almost always know at which part of the Roman Empire you are, chronologically-speaking. Also, later roman coins were minted during a period where there was an elaborate mint-mark system. This means that not only can you attribute the coin much easier, you can even attribute it right down to the mint itself (and even further!!!)
Another advantage is that Roman coins are generally found in higher grades than Greek coins. A quick search on your favourite coin sites, or even indeed eBay will readily confirm this.
Another advatange, though this is a moot point, is that it is easier to determine whether a Roman coin is authentic or not. This is because generally-speaking, Roman coins feature a known and familiar styles, thus easily identifiable to the collector. Regarding later issues, the design becomes so simplistic that any deviations from the original style are pretty evident to the average collector.
Compared to Greek coins, Roman coins are not as diverse in styles and types. Also, arguably speaking, Greek coins on average exhibit greater artistic style. Certainly the classical period features coins that near perfection.
Another disadvantage is that many emperors are simply beyond the collectors reach, if not for their incredible expense, then for their immense rarity. For some emperors/empresses, a collector may have to wait years before one becomes available.
An immediate advantage to the collector is that he/she has much scope for collecting as there is a more diverse range of coin types availalble.
Also Greek coins come in the most fascinating variety of shapes and forms. A collector on a tight budge may survive by purchasing common coins such as obols fromcommon areas and/or mints. However, unlike with Roman coins, that does not mean that the collector will never come accross coins with exquisite designs. In fact, most greek coins contain amazing depictions of mythological scenes and/or gods, goddessess, kings and queens. This is quite different from the standard Emperor Obverse and standard (for common coins anyway) reverses for the coins of the Roman Empire.
The extraordinary range of coin types availabe makes it extremely difficult for the collector to frame his collection.
Also, Greek coins tend to get expensive quite quickly. If you are collecting bronze coins, you will delight in the sheer number of coins available to you for only a few dollars. However, if you progress to silver coins, you will quickly find that some coins will never be in your price range.
Like Roman Coins, Byzantine coins are cheaper than greek coins. In particular, Byzantine cup coins in VF condition can be foundeBay for as little as $US 2.00 each. Gold and silver coins are also generally cheaper than the average Roman Solidus or Aureus.
To the untrained eye, Byzantine coins of the same type are virtually indistinguishable from one another. This makes attribution an extremely difficult process.
Also, the style is cruder, and do not match the celatorship found in the Greek and Roman Empires.
The main advantage is that at face value, they are cheaper. This should not be the main consideration of the collector, unless he/she is working with an extremely limited budget, because it is an often misleading point. The statement does not take into accound the opportunity cost of cleaning coins. Cleaning coins takes time and patience and in those many hours, the collector may have instead chosen to work a few extra hours to increase his/her wages. However, there are collectors who manage to efficiently refine their cleaning techniques and thus decrease this opportunity cost.
Another advantage though, is that by cleaning and attributing the coins, the collector finds that he/she is able to learn a great deal about ancient numismatics and history in and fun and enoyable way.
A fundamental advantage to collecting uncleaned coins however, is that there is a chance that the collector may stumble across rare and even unique types. It is not uncommon for a collector to find an unpublished coin type in a large uncleaned lot.
Contrary to any claims by dealers, uncleaned coins are "cherry-picked" 100% of the time. The coins have passed through several hands usually, before they land in the collector's.
Also, larger uncleaned coins are almost never found in high grades in uncleaned lots.