Ancient Rome was heavily influenced by ancient Greece in its architecture, literature, education, and language.

Greek architecture was famous for its harmony, proportion, and use of columns. These ideals had an enormous influence on Rome. For instance, the Pantheon, the best-preserved Roman building, demonstrates the importance of Greek thought. The word Pantheon is Greek for "honor to the Gods." Apollodorus of Damascus, a renowned Greek architect, helped design it.

Greek literature also had a profound impact on Rome. The Greeks founded poetry, history, and other genres of literature. The Iliad and the Odyssey impressed Romans greatly. Lucius Livius Andronicus, the father of Latin literature, was a Greek-speaker and slave who translated the Odyssey into Latin. Virgil, the author of the Aeneid and the dominant figure of Latin literature, was shaped by Greek culture.

Enslaved Greeks taught Romans. Rome accepted the Greek ideal of education and added more emphasis on public speaking. Roman senators were often bilingual in Latin and Greek.

Yes, the Romans were great borrowers and assimilators, which contributed to their success at empire building.

The overwhelming cultural influence on the Romans came from Greece. The Romans adopted the Greek religion, Greek art and architecture, and Greek education, emulating that culture as the most "sophisticated" they knew of, much as the Japanese adopted Chinese culture.

They supposedly learned to trade and the concept of building impressive cities from the Etruscans, a group that lived just north of them in the area now known as Tuscany. They also borrowed the idea of having gladiatorial combats from the Etruscans and learned how to drain marshes from them.

They borrowed gods from Egypt as well as Greece. The Romans, for example, adopted Isis, an Egyptian goddess, and constructed a temple to her in Rome. According to the historian Josephus, Isis was widely worshipped throughout the Roman Empire.

Roman culture is distinguished by numerous borrowings. Although it did innovate in engineering, admninistration, and law, much of its culture was derivative.

Roman religion was originally influenced by Etruscan and then by Greek religion. The Roman gods slowly became assimilated to the Olympian pantheon, with Jupiter becoming a Roman Zeus, Juno a Roman Hera, Venus a Roman Aphrodite, and so on.

The Roman system of roads and postal services for government business followed earlier Persian models. Much of the Roman empire's structure followed earlier Persian and Hellenistic models. Roman literature overtly imitated Greek literature. The famous Roman comedies of Plautus and Terence were actually adaptations of Greek originals. Virgil's Aeneid was a continuation of Homer's Iliad. Much of Roman philosophy followed older Greek models. Many Roman sculptures were copies or imitations of Greek originals, and some aspects of Roman architecture, such as the use of columns, were based on Greek originals.

Rome absolutely did borrow things other than styles of clothing from other cultures.  Most notably, Rome borrowed many things from the Greeks.  They also borrowed some aspects of other cultures. 

Perhaps the most famous way in which the Romans borrowed from others was in religion.  Roman religion was heavily influenced by Greek religion.  This influence came to them through the Etruscans.  The Etruscans adopted the Greek gods and those gods were then adopted by the Romans.  The Romans came to have equivalents to all the major Greek gods (Jupiter for Zeus, Mercury for Hermes, etc.).  Later on, the Romans also came to borrow religions from other cultures.  Romans came to adopt religions from such places as Egypt and Persia.  In these ways, they were borrowing from others.

Rome also borrowed heavily from the Greeks in terms of education.  Roman children (of the elite classes) were often taught by Greek slaves.  The things that they learned were patterned after what the Greeks thought important in education.

The Romans also borrowed from the Greeks in the arts.  They translated Greek plays into Latin.  When they wrote in Latin, they often followed Greek traditions and plots.  They also adopted Greek styles in things like painting and sculpture.

In short, the Romans borrowed liberally from other cultures, particularly the Greeks.