There are no nickels or dimes series here to collect, so, the answer is whatever interests you! This is where ancient coins become really fun; you pick what you want to collect,and you can became a small expert in the field.
Many collectors of Roman coins try to obtain coins with portraits of as many emperors as they can. While this will make a historically interesting set, you will not be able to get them all. Some emperors kept their heads for too short a time to mint enough coins to make them affordable (or even obtainable). However, you can get about 80% of the emperors or so, depending on your budget. If you do decide to collect by portrait, I would suggest that you also try to pick a theme for the reverses of your coins to make them more interesting.
While a collection of emperors is very worthwhile, you may want to be able to say something about what is on the reverse of your coins or have some interest in it. You may also want to stay away from coins that have a very clear image of the emperor but the reverse is not clear at all or worn down.
There are numerous possibilities for collection themes. Some themes that you may want to consider are gods (Mars and Jupiter are common) and goddesses (Venus or Juno for example), personifications (Hope, Loyalty, Victory…), animals (eagles, horses or lions are common), military or weapons (armor, shields, spears, standards, trophies, captives or legends referring to conquests), temples or Christian symbols (such as the Christogram or a cross).
Issues of a specific emperor who was around for a while and minted many coins would be meaningful, possibly giving you insight into his life or the period in the Empire’s history.
You could specialize in a specific denomination, such as the nice 3rd century silver Antoniniani, or the smaller AE (bronze) coins of the 4th century, or in a given century – the little 5th century bronze coins look very different from the large 1st century bronzes. Mint marks would make a great geographical collection – trying to get coins from all the mints of the empire, or maybe many different varieties from just one mint. Perhaps you were born in one of the cities where Rome minted coins and would like to collect coins issued there. Many emperors had coins issued while they were still caesars, and also coins of their wifes which tend to contain different reverse designs.
Posthumous issues made after emperor’s death are unique, as are legionary issues which were used to pay the army (minted by Septimius Severus, Gallienus, Victorinus and Carausius, as well as Mark Anthony). Historical issues made at times of famous battles or events, such as issues with titles like BRIT, GERM or PART, denoting the emperor’s conquest and victory are very neat. There are coins for the 1000th (under Philip I and II) and 1100th (under Constantius II and his brothers) anniversary of Rome, as well as issues commemorating the founding of the city of Constantinople.
You could concentrate on specific reverse legends or designs, and even on errors such as spelling, missing letters or double strikes. Coins of emperors wearing radiate crowns would look cool, as would a collection of Provincial issues or Barbarous imitations. Even counterfeit coins (both ancient and modern) would made an interesting collection! As you see, there are many possibilities.
If you only want to spend a few bucks ($5-10 a coin) here and there, you could collect one of the common 4th century emperors, such as Constantine the Great, any of his four sons (Crispus, Constantine II, Constantius II and Constans), or Valentinian I, Valens and Gratian – many of the above themes will be possible here. The military emperors Probus or Aurelian issued very many interesting coins that start in this price range.
For just a bit more you could collect the nice silver pieces of Gordian III, Philip I or II. Maybe the emperors from the Gallo-Roman empire will interest you (Postumus, Victorinus, Tetricus I and II are readily available).
There are many nice and interesting types available for all of these and more, some more common than others of course, and each one is a possibility for the main theme of any collection. Furthermore, they can be mixed – military issues of 3rd century emperors, silver coins with gods or goddesses, coins from all the mints used by Constantine the Great, etc…
You might want all your coins to be nice and clean, or you might like them
a bit dirty and damaged so that they are truly ancient looking. Consider the coin in Figure 3 below; don’t the chips around the edges look neat? Can you make out the legends? If you can’t, then image being able to – it’s like doing a bit of archeology by yourself, trying to decipher an ancient text. With some practice reading such coins becomes very easy, and just by its style you’ll be able to narrow it down to late 4th century. Every coin will contain interesting aspects which can make it very collectible by collectors with various interests.