Follow up question for OP : are you referring to the literacy rate or to the actual physical access of common citizens to written texts/works ?

Common people would encounter writing on a daily basis. The Romans put inscriptions on virtually all monuments, tombstones and coins. Just like in modern days there was graffiti on walls, including election slogans.

 

Inscriptions

The Corpus Inscriptionum Latinarum is an almost complete collection of surviving ancient inscriptions in Latin. You can access it online!

Here is an example of such an inscription on the Pantheon, one of Rome's most famous monuments. The text means "Marcus Agrippa, son of Lucius, made this when consul for the third time". As you can see the Romans were used to abbreviations. In this example "M" means "Marcus", "L" means "Lucius", "F" means "filius", and "COS" means "consul".

Apart from dedicatory inscriptions on monuments the average Roman would also encounter inscriptions on tombstones. These are so common that virtually every museum with a Roman department has some on display.

 

Coins

Roman coins usually have inscriptions on them. These too use many abbreviations. This website has a nice collection of pictures. Common abbreviates include the ones mentioned above, and also things like "IMP" for "Imperator", "PM" for "Pontifex Maximus", and "AUG" for "Augustus".

 

Books

These were a bit harder to come by. Publishing a book basically meant an author completing the manuscript and sending it to friends for them to read. Anyone who wanted a copy would set scribes to work. Otherwise it was common among aristocrats to borrow books from one another. Books largely spread through these kinds of informal networks. There was a commercial trade in books too, and imperial Rome actually had bookshops.

There were public libraries from the first century BCE onwards. The first were founded by Caesar and Augustus. In theory these were actually open to the public, but to what extent they actually were is debated. We're sure that the upper classes and artists under their patronage had access to the public libraries.