Romans went shopping to the market which was located in the Roman Forum. In the early days, the Forum was a swamp which the Romans drained in order to build the first paved street: the Via Sacra. It had the first government building called the comitium and the rest of the open square had the market which contained many shops.
By the end of the Republic, the market was moved to make more space for other government buildings, temples and beautiful monuments. During the Empire, many other forums (fora in Latin) were built for example: the forum of Augustus, Vespasian, Nerva, Trajan, etc.. noting that the forum of Trajan can still be seen today. It is worth noting that the streets leading to the Forum and the Via Sacra itself also had many shops.
There were other markets throughout the city of Rome such as the market of the subura quarter for the poor Romans which mostly sold vegetables and chickens and which was mostly frequented by slaves. There were also neighborhoods with luxury shops such as the ones found in the Campus Martius.
The Roman Forum was bustling with people noting that carriages were prohibited during the day. It was full of pedestrians and wealthy Romans being carried in litters. During the Empire, all nationalities could be seen in the Forum and all kinds of products could be purchased including products from far away lands.
The market was huge. It contained many kinds of shops, shops that sold food, spices, shoes, wool, books, etc. There were barber shops, blacksmiths, etc. The forum boarium right next to the main market was a huge meat and cattle market while the forum cuppedinis sold luxury goods.
The Forum was more than a market. It was a place of business kind of like downtown in American cities. Businessmen made deals in the basilica (the multi-story building that housed the shops). Money-changers or the argentarii who worked at the tabernae argentariae which were the equivalent of banks exchanged coins (foreign coins to Roman ones).
They also held money and paid interest just like banks do today. And just like banks today they used that money in other profitable transactions or investments and acted as agents. During the Empire, the argentarii would also circulate newly minted coins, again just like banks today! It is worth noting that at some point the tabernae argentariae were moved to the forum boarium while the cattle market was moved elsewhere. There is still an arch that can be seen today that symbolizes the guild of the money-changers.
Interesting facts about ancient Roman shopping