Short answer, yes. Very much so. The Romans were quite flexible when it came to military equipment and tactics, constantly modifying their system based on their current opponents.

Let's start in the 4th and 5th century B.C. Contact with the hellenized Etruscans likely introduced the Romans to hoplites and the methods of phalanx warfare, which they adopted. Ivory plaques in the town of palestrina in latium show men in close formation with hoplite equipment. With phalanx combat, it is likely the Romans were equipped similarly to hoplites at this time.

At a later time, the Romans abandoned the phalanx in favor of a looser tactical deployment based around smaller units called maniples. Although hard to date , the transition to maniples occurred in the 4th century B.C. It is not clear if maniples are a Roman invention or if they borrowed them from someone else. Equipment is still quite varied at this time and it is not totally clear if equipment was standardized until the mid-republic.

Our best source for this mid republican era is Polybius, who wrote around 150 B.C. Polybius descibes some of the equipment used by the legions and the archeological record mostly supports this although not entirely. When the Romans started using maniples as their fighting units, they also adopted a formation called the acies triplex (tripartite battle formation) or 3 lines. The Romans did not always deploy in three lines but to make things simple I will cover the 3 types of infantry laid out in the sources.

Hastati were young men, in their late teens and twenties. They occupied the first and sometimes second lines of the acies triplex formation. Equipped with rectangular convex shields called the scutum. About 4ft high and 2 ft wide, made of wood and hide with a metal boss or hand guard in the middle. The top and bottom edges were reinforced with iron. Simple Helmets were worn with cheeck pieces for face protection and feathers to add height. Pectorali were worn on the chest as a form of simple breastplate to protect the vitals. More affluent troops could afford chainmail shirts. Only one greave was worn at this time, on the left leg. Hastati carried 2 pilum that were used as javelins or sometimes (albeit more rarely) spears. The Hastati also carried the Gladius Hispaniensis or spanish sword, likely copied from the Carthaginians or Spanish mercenaries fighting with the Carthageinians during the punic wars. The blade is about 1 and a half to 2 feet long.

The Principes were lined up after the Hastati, generally in the second line and were a little older although equipped the same. Triarii were armed like a hastatus or princeps, only with a spear (hasta) instead of the pilum. These were the oldest and most experienced troops in the Roman legion.

The Later Roman army was quite different from that of the preceding Principate and of the Roman republic. Changes came piecemeal through Marius, Gallienus and others. Sources include Notitia Dignitatum (A.D. 390), Ammianus Marcellinus (mid-4th century), and Vegetius (A.D. 400).

Differences are clear in a number of categories. Legions became smaller but more numerous. There were more small-scale actions and fewer major engagements. Tactical organization also changed at this period in response to greater external threats to the Empire. Emperor Constantine divided into garrison troops (limitanei) and field armies (comitatenses), somewhere around A.D. 300-340. In addition to organizational changes, equipment was changed to support the more defensive nature of the Roman army at this time.

Distinction between auxiliary and legionary vanished, most were armed the same. Helmets, body armor and greaves become less common. Shields changes from the rectangular scutum to the flat, oval shield, five excellently persevered examples survive at dura-europos an outpost on the Euphrates river. The Spatha, a longer sword, replaced the Gladius. Originally used by cavalry and more familiar to the germanic recruits more commonly used to fill the ranks by the 4th and 5th centuries A.D. Pilums were replaced with long stabbing spears. Darts were also used, 3-4 feet in length, sometimes clipped onto the inside of the shield.

Edit for Sources:

Ammianus Marcellinus, The Later Roman Empire

Connolly, Greece and Rome at War

Lenski, Failure of Empire

Keppie, The Making of the Roman Army

Livy, The History of Rome

Polybius, The Rise of the Roman Empire

some more edits for grammar and clarity

another edit based on /u/Rittermeister's question