The Romans came into contact with Gauls and other Celts around 400BC [1]. At the time, various tribes of Celts occupied large territories from Anatolia in the East to Britain and Iberia in the west. [2]
At the time, the Romans were basically using the same weapons and tactics as the Greeks, but they had trouble when fighting the Celts when the Celts managed to breech their lines.
In 390 BC, the celts sacked Rome [3]. In response, Rome was fortified and the the Romans would be training and organizing in ways intended to withstand the Celtic charge. Most importantly, that meant improving cohesion and discipline to the point where they could hold the line.
Gradually, though, the Romans would also incorporate Celtic arms and armor, both from Gaulic Celts and Iberian Celts, perhaps most famously the Gladius [4].
Eventually, Rome started to push the Celts back, but there were setbacks, such as when Hannibal employed large numbers of Celtic soldiers during the Punic Wars [5].
Eventually, the Roman military came out if these conflict as a much stronger force when it had started, and ended up as a force that was even able to match the Macedonian phalanxes (partly because the Macedonians did not employ cavalry with the skill that Alexander had).
While Rome was busy conquering the Mediterranean, Germanic tribes were pressuring the Celtic tribes that had occupied modern day Germany, partly replacing them and partly subjugating them.
The Cimbrian war started in 113BC (the Cimbrians themselves may have been Germanic or Celtic, but the Teutons supporting them were definitely Germanic, and the German name for Germany (Deutschland) is taken from them).
Initially, the barbarians were on the offensive, but in the end the Romans under the leadership of Gaius Marius were able to push them back.
In response to the weakness against the Germanics, Marius would institute his "Marius Reforms", replacing the citizen army with a much more capable professional army. [7]. These soldiers are the most famous incarnation of the Roman Legions, armed with Pila, Scuta and and Gladii.
Those reforms, incidentally, paved they way for the Fall of the Republic.
[1]. https://en.wikipedia.org/wiki/Cisalpine_Gaul
[2]. https://en.wikipedia.org/wiki/Celts
[3]. https://en.wikipedia.org/wiki/Battle_of_the_Allia#Aftermath,...
[4]. https://en.wikipedia.org/wiki/Gladius
[5]. https://en.wikipedia.org/wiki/Second_Punic_War
[6]. https://en.wikipedia.org/wiki/Cimbrian_War
[7]. https://en.wikipedia.org/wiki/Marian_reforms