I can only really provide an answer in terms of the Roman elite of the 3rd-2nd century BC, but you are correct that socio-economic factors definite played a part. Your average Roman citizen in Rome would have most likely only known Latin (and most likely have been illiterate) - however, for those who were educated they would have most likely have known Latin and Greek.

The influence of Greek language within Rome is seen even before it became the language of the Mediterranean (following the conquests of Alexander). A temple of Ceres, Liber, and Libera was built in 493 BC (Dionysius of Halicarnassus, Roman Antiquities. 6.1-2) which was allegedly influenced by the Greek deities Demeter, Bacchus, and Kore and according to Pliny, was decorated by the Greek artists Damophilius and Gorgasos (Natural History. 35.154). The rituals of such temple were of a Greek origin and performed by Greek priestesses (Cicero. For Balbus. 55), and thus most likely in the Greek language. There are examples of Greek cognomina being taken by those within the Roman elite during the fourth and third century, some achieving consular status such as Philo (339), Sophus (304), Philippus (281), and Philus (223).

There are more obvious examples of the Greek language within Rome. Some Romans, such as Cato and Varro, believed Latin derived from Aeolic Greek (Dionysius of Halicarnassus, Roman Antiquities. 1.90.1; Varro. On the Latin Language. 5.25-26, 96, 101-102). An infamous episode is that of L. Postumius Megellus who carried senatorial demands to the Tarentines in 281 and delivered them himself in Greek only to be jeered for his pronunciation and kicked out of the theatre, resulting in war between the two (Dionysius of Halicarnassus, Roman Antiquities. 19.5; Appian. Samnite Wars. 7.2); Latin became the only language in which Roman official state business was conducted from this point onwards (Valerius Maximus. 2.2.2).

There are several written examples of members of the Roman elite being well versed in Greek. Plutarch states Flamininus was incredibly skilled with the Greek language (Flam. 5.5), negotiating and conversing without an interpreter (Plutarch. Flamininus. 6; Polybius. 18.1, 18.4-9, 18.37). Aemilius Paullus conversed freely in both Latin and Greek without hesitation (Livy. 45.8.3-7; Valerius Maximus. 5.1.8). Cato was able to speak Greek in an official capacity (Plutarch. Cat. Mai. 12.4) and had the knowledge and ability to recall and quote Greek literature and myths (Plutarch. Cat. Mai. 8.2-3, 9.3, 23.2, 2.4) - so, someone who is known for their dislike of Hellenistic culture was still very well versed in the language. In fact, it is said he believed engagement with Greek literature would corrupt the Roman mind (Plutarch. Cat. Mai. 23.2; Pliny. Natural History. 29.14) which implies Greek literature could be read.

In 181 BC, books of Pythagoras were burned implying that Greek literature was present within Rome (Liv. 40.29.3-14; Valerius Maximus. 1.1.12; Pliny. Natural History. 13.84-88; Plutarch. Numa. 22.2-5). Livius Andronicus, active during the third century, translated the Odyssey from Greek to Latin, teaching in both languages (Suetonius. On Grammarians. 1.2). Likewise, Fabius Pictor composed a history in the language of Greek. Further evidence that Greek was a language for the literate. In short, Greek was very common among the educated.

The significant import of slaves from Greece to Rome following the conquests of the second century BC does raise questions regarding the usage of Greek among the common people, and it is a very possibility that within Rome there existed those who could speak Greek and Latin, but that is something I am not entirely sure about.

In terms of other languages, there is an argument to be made that Punic was known during the period. Poenulus by Plautus contains instances of Punic language (lines 930-939). Likewise, it is said that the writings of Mago were translated into Latin by D. Junius Silanus and Greek by Cassius Dionysius during the second century BC. One would suggest that such that there was knowledge of the Punic language within Rome.

Hopefully this can give you an insight into your question. If you would like to understand further into the question, I would recommend to begin with:

J.N. Adams (2003), Bilingualism and the Latin Language.