I find this period history ( the Roman period) very interesting but have never spent any significant time studying it, only bits here and there.  I was under the impression that Rome had fallen somewhere in the 3rd or 4th century AD but this battle took place almost 400 years before that.  
Battle of Allia

Closer to 800 years than not.

Don't forget that Rome was not always the strong power it wound up being--- it took a lot of time and effort to build it up.

Rome got sacked more than a few times, the 400 ad (to the Romans) was just another one.

Rome was originally a city state. Similar to how the Greek city states (polis), Athens, Sparta, Corinth, Thebes, Rome ended up dominating its neighbors, other Latin city states, and non-Latin Italians including the Etruscans and the Samnites. But Rome managed to consolidate its power more effectively, while the powerful Greek city states were forced to use various types of Leagues, which they subtly controlled, Rome dominated its neighbors and either absorbed them, making them Roman too, or tied them to themselves as Socii allies (Friend and Ally of Rome), through an alliance system of protection and mutual security. They didn't have a Peloponnesian League, or a Delian League, they simply had the Roman Senate and People, everyone else was subordinate, lesser citizens in the whole, but given autonomy to govern their own domestic affairs.

During its wars with the Etruscan city states, Gallic tribes of the north Italy region, in the Po Valley, were used as quasi mercenaries. When Rome defeated the Etruscans, taking control of the lands of northern Italy on both sides of the Apennine Mountains, it demonstrated to the Gallic tribes that Rome had its mind on expansion. In retaliation, a large confederation of Gallic tribes formed up and marched south, defeating the Roman field army at Alia, and sacking a good part of the Rome itself, while the remnants barricaded themselves on the Capitoline Hill. At that point, most Romans didn't live in Rome, they lived in outside towns and villages, so only a small portion of the total forces were actually stuck inside the city. While a relief army was raise, the Gauls eventually pulled out and themselves were defeated by the famous Cincinnatus. Following the war, Rome reformed its military to make it more effective, and strengthened the city walls, as well as embedding military colonies further north, to protect the frontier from further Gallic invasions. Within a 150 years, all of Italian Gaul and Southern Italia (Magna Grecia/Greater Greece) was conquered, giving Rome total control of Italy.

Correct me if I'm wrong, but wasn't it at this time that the Roman army underwent the change from a phalanx model system to the manipular system?

Yes, around that time period but sources are conflicting a bit, some say Allia was the reason for the reforms, while other sources say the 1st and 2nd Samnite War (fought in mountaneous areas), fought in the 4th-3rd century BC, were the reason for the switch to the manipular legion (decentralized infantry lines, subunits controlling their own battl space semi-independently of one another, with reserve lives).

Personally I believe that the reforms occurred over a long period. Previously, the elite of Roman society were hoplite equipped, after one reform they became cavalry. Cavalry was probably emphasized after fighting the Gauls at Allia, as most Gallic tribes produced great cavalryman (their nobility fought mounted).

While the Roman elite had shared some panoply with the Greeks of Italy and Hellas, specifically the clipeus/aspis shield, they still likely deployed a bit differently than well known Hellenic city states and weren't uniform in their use of any types of equipment. While the 1st class was still carrying Hellenic panoply the 2nd-5th Classes likely carried scutum/thureos style shields before Allia, as there is iconic imagery and archaeological finds suggesting they had been used since the 7th century BC.

Most of the great reforms of the Romans have been oversimplified by historians. The Camilian Reforms are notorious, as are the Marian Reforms.