What gods were foreign and then adopted by the Greeks and Romans through trade/war/etc?

A few of the most famous examples of Roman religious syncretism are the cults of Isis, Mithras, and Sulis.

Isis was an Egyptian goddess of magic and mother of Horus, God of the Pharaoh. She was an incredibly important figure in the Egyptian pantheon so it's no small wonder that her cult thrived under the rule of the Ptolemies after Egypt was conquered by Alexander of Macedon in the 4th century BCE. When Cleopatra died and Octavian claimed Roman conquest over Egypt, slowly the cult of Isis began to take hold. Her cult would later become wildly popular amongst the Romans in Egypt, who would spread it far and wide. A large temple to Isis was even found as far north as Londinium, thousands of miles from her original home on the banks of the Nile.

Speaking of Roman Britain, one of the Romans most effective methods of conquest was extremely passive, that is to say they imposed no religious doctrine on their recently conquered peoples. Rather, they attempted to find common grounds between the native deities and their own. A god called Sulis had a large following within the north of Roman Britain, near what is currently the city of Bath. Sulis shared many properties with the Roman goddess of war and wisdom Minerva, so the Romans stuck them together and called the deity Sulis Minerva. In this way not only did the Romans assuage any fears of Roman imposition of religious practice, but they also helped introduce those practices passively. Scholars have argued this was almost a more effective tactic than military might for ensuring conquered peoples stayed Roman (although the issue is far more complicated than that).

Speaking of the military, the cult of Mithras was hugely popular with them, and with the military being found in all corners of the Empire, it's no surprise that cults to Mithras are also found everywhere in the Empire. Mithras was originally a Zoroastrian deity from the near East who was implanted throughout the empire by those same soldiers, and the cult spread from there. Mithraism is a highly structured religion, with seven levels of religious initiation. This highly rigid rank structure mirrored the rank structure the soldiers were already familiar with in the Army and provided yet more incentive for promotion (one has to wonder if an experienced Mithraic infantryman had the potential to exercise power over his newly initiated commanding officer). Mithras was also a warrior and sun deity, the symbolism and significance of which also mirrored the desires and lifestyles of the Roman military.