The first major battle for Rome becoming such a major power would've been Zama in 202BC, where the Romans effectively crippled the other super power in their region at the time, Carthage. This early battle in Rome's history made it the most dominant power in the Mediterranean giving Rome the opportunity to turn its look beyond the Mediterranean and continue its expansion in other directions. Taking Carthage would also give the Romans a foothold in North Africa

Other notable battles could consist of Pydna 168BC, where the Romans reinforced their control over Greece, which would give them a foothold to later expand into southern Dacia, Illyria and into modern day Turkey

Alesia in 58BC which saw Julius Caesar face off 200,000 gauls with his 60,000 legionaries. With the victory of Alesia, Caesar was able to subdue Gaul as a Roman Province after an 8 year campaign (Roughly modern day France). The confidence and popularity Caesar would gain here would lead him to march on Rome with the collapse of the First triumvirate and start a civil war that would see the end of the Roman Republic

The next and very immediate battle would be Pharsalus 48BC where caesar would defeat his rival, Pompey the great, effectively ending the Republic right there and then

The Battle of the Nile 47 BC would see Caesar end the civil war in Egypt securing Roman interests in the middle east (and the Egyptian throne) giving them a foothold to later establish client states in the area

Again, very immediately, would the Naval battle, Actium 31BC also be of significance. This battle saw Octavian cripple Mark Antonies forces outside Greece, routing Antony in the process. This saw Antony eliminated as a political opponent and would leave Gaius Octavian to become the first Roman emperor who would lead Rome into the period known as the "Pax Romana" a period of time that Rome would flourish and prosper

(So by this point Rome controlled Italy, Magna Graciea, Greece, Hispania, North Africa, Egypt, Illyria and western Turkey)

Battle of the Medway 43 AD would see the successful invasion of Britain by Rome

Second Battle of Tapae in 101 AD would see Rome control the province of Dacia

After this point there were probably a few more major Roman victories in major battles but apart from that Rome just slowly continued to expand until a point was reached that it eventually began to secede from its set borders with the increased barbarian inquisitions. Famous defeats of Romans such as Cannae, Teutoberg forest and Adrianople were just a few examples that defied the idea of Roman invincibility

Edit me if I forgot or got something wrong