There were no standardized symbols for addition and subtraction until long after the Empire. The symbols we use now date from after the 15th century.

For the most part, computations (even quite complicated ones) would be written out in words. For an example of this, you can look at Frontinus' (1st century AD) De aquis (recent translation here), where he discusses the water supply system in Rome in detail. Frontinus' surprisingly intricate calculations have been discussed at length by many authors (see the references below).

Of course, many of the famous Latin works we have are presumably somewhat polished, and present mostly final results, without detailing the computations that lead to them. Unfortunately, we do not have many primary sources explaining how the Greeks or Romans performed computations. Most of what we believe is based on guesswork from examining authors such as Frontinus or Diophantus. The classical reference for this is Friedlein's Zahlzeichen und das Elementare Rechnen der Griechen und Roemer (Numerals and elementary arithmetic of the Greeks and Romans). You can look at the bibliography in this article by Maher and Makowski for a more up-to-date overview of this question, and a detailed discussion of Frontinus' computations. We do know that the Romans used abaci, which, if operated expertly, can certainly rival computation with pen, paper, and modern notation, and learned multiplication tables.

Even in the few examples of more informal computations we have, (Serafina Cuomo gives examples from Greek papyri of the Roman period in her book Ancient Mathematics), computations were generally written out in words. In A History of Mathematical Notations, Cajori examines the example of Diophantus (who lived in Alexandria under the Empire). He did have his own notation for subtraction (an inverted trident or "psi"), but he would just put numbers next to each other to signify addition. This notation was not widespread, and did not remain in use after Diophantus. Cajori also mentions "sporadic" use of a slash (/) to indicate addition in Greek papyri of the period.