Most beginners have trouble knowing what to do once they've cleaned their first coin. If you're not sure what to do either here are a few pointers.
Just about any ancient Roman coin you are bound to find will have the name of the emperor (or that of his relative) written out on the portrait side of the coin. The easiest way to identify your coin is by using the clues provided by this lettering. You do not have to be able to read all of the letters but the more that you are able to pick out the more focused your search will become. When it comes to Roman coins, attributing your find successfully is more often than not an elimination process that leaves only one, or few, possible rulers to whom it can be attributed.
Unfortunately, you will find that many of the coins that you clean will have all or parts of the lettering worn off or unreadable for any number of reasons. While it is usually possible to ID a coin with no lettering remaining it undoubtedly becomes a more challenging task. Your best bet is to use the clues provided by the remaining details.
To begin, let's analyze a typical coin of the type you are likely to find:

In this case, our coin has distinct lettering on both sides but the individual letters are a bit blurry. For certain on the "heads" side of the coin we can tell for sure that on one side it looks like STAN and by his forehead TIV. When we enter the string STANTIV into the database's search engine four entries are returned; all of which seem to be variations of CONSTANTIVS. Suddenly, we've narrowed down the possibilities from about 200 to just four. In hindsight, the rest of the lettering to the left side should become more readable now that we're "forced" to see a CON before the STAN.
More about Roman writing oddities
But this is not acceptable. We would not want to look through literally hundreds of entries on the database for a match under all four rulers. How can we narrow down the search even more? At this point we should look to the "tails" side of the coin for additional clues.
It should be immediately obvious in this case that there seems to be a struggle between two people, the luckless one on the bottom receiving a fatal blow with a spear. Reading again the letters the clearest ones seem to be the second through fifth bearing out ELTE. Returning again to the search engine, it seems that this letter sequence happens for the coins of only six rulers. Of those six only two were hits in the previous search, Constantius II and Constantius Gallus. So we have now narrowed it down to just these two. If we were to search within each page for the ELTE sequence we would quickly see that both emperors have an affinity for the very same reverse of type FELTEMPRAPARATIO which are indistinguishable in design from one to the other. So is this as far as we can go? No, the portraits now provide us with important distinguishing features. Notice that all of the Fel Temps of Constantius Gallus have his portrait with a bare head while those of Constantius II have him with a pearled headband. The lack of this headband, called a diadem, is symbolic of a ruler who had not yet become emperor but was merely a prince. This is also borne out under careful examination of the writing which for Gallus ends in NOBC while for the emperor Constantius II they end in a more explicit AVG for Augustus, the formal title of an emperor.
We have therefore determined that this coin is a Fel Temp Reparatio of Constantius II. However, if we wanted to we could go even further thanks to the last bit of information which is clear to us. The lettering of the exergue. The exergue is that portion of the reverse side of a coin directly at its six o'clock and usually under a line. In this area go various mint marks similar to the solitary letters we find in our modern age pocket change which tell us where the coin was made. In the case of Roman coins, exergual markings often became very complex and difficult to understand. They were not written for the benefit of the public which used them but rather for administrative bodies which needed to track the flow and quality of the coins from all over an enormous empire. In the case of this coin we are fortunate to be able to read very clearly CONS with another partial letter that cannot be made out. Browsing over the list of dozens of different coins for Constantius II we are overwhelmed to find many coins which not only come from the mint of Constantinopolis but also just so happen to be Fel Temp Reparatios as well! Looking a little closer, however, we can see subtle differences in the mint marks and/or legends (the lettering we decoded). Because numismatic historians have carefully studied each minor variation, they have classified sometimes hundreds of varieties for a single ruler with a single type of reverse. Although initially daunting, it should become trivial to find the combination that matches closest to the coin you have to get the correct attribution. However, today we're lucky and we see that the entry for RIC 119 matches perfectly. The fragmentary letter after CONS, it turns out, is an officina. This is an internal department within the actual mint building equivalent to saying "...And this coin came from the 5th floor of our mint facility".
Because of this great variety of subtle differences, you will likely come across coins which do not have an exact match on the image database. Don't despair if this happens, there are thousands of coins and no image database is likely to have every combination. It is also a little "special" to get a coin not matching entirely as this means it must not be very common. What should be most important in your attribution quest should be to match your coin to the correct emperor. Getting the final attribution as to exact catalog number (if not found here or on another database) will require access to that reference book. In the case of RIC (the Roman coin reference "bible") the entire series runs into about a dozen books costing over $1,000!
It is our hope that this primer will have given you the critical first few steps on how to begin with your first coins once you have cleaned them. With practice it will only take you a few glimpses of a coin to know whose emperor it belongs to!