I know very little about Iron Age Gallic warfare, so I won't comment on the ancient period, except to say that Vercingetorix originated from the modern-day Auvergne (he was the leader of the Arverni tribe from whom the region gets its name) in south-central France, which has its capital near modern-day Clermont-Ferrand. In contrast, Charlemagne was an Austrasian nobleman, whose heartland was centred between the Marne and Rhine rivers (especially the Moselle valley) and had his (eventual) capital at Aachen in modern-day Germany. So although both are claimed historically by French national myth-makers, their contexts were quite different, separated as they were by c. 800 km.

Anyway, on to Carolingian warfare. In theory, under Charlemagne each free adult male was supposed to assemble each spring to join the royal host, where an assembly would be held, political decisions made and the annual campaign would take place. The local count (aided by the bishop - the two roles were seen as complementary) was in charge of assembling the men of his county and bringing them to the royal host. In reality, those who attended and took part in Carolingian politics and warfare (particularly in the early part of Charlemagne's reign, there was little distinction between the two) would have been drawn from the wealthier portions of society and we know from surviving capitularies (proclamations made at councils) that it had become acceptable for two or three free men to band together and pay for one of their number to go fight on their behalf. Although there was certainly some involvement from non-noble soldiers (a practice that was slowly dying out), the leadership would have been purely with local aristocrats and a decent proportion would have been made up of them and their personal household troops. In particular, a small group of immensely wealthy families, called the Reichsaristokratie ("Imperial Aristocracy") by historians, were particularly prominent in warfare and politics and would have contributed large numbers of troops. Monasteries and other churches which benefited from royal patronage would also have been a major source of troops (either directly from the monastery's lands or equipped and hired using the monastery's resources). These weren't professionals per se, but warfare was well on its way to becoming something that only trained and well-equipped men (usually from the aristocratic classes, who could afford to equip themselves and practice) participated in.

As for weapons and tactics, we know very little. Like much of the rest of warfare in the middle ages, pitched battles were rare, and much of the fighting seems to be small-scale skirmishes and raids, intended to disrupt enemies' productive capacities or to seize movable property and captives - plunder was the raison d'etre for many Carolingian campaigns. The basic requirement for a soldier, IIRC, was a spear, shield and helmet, but we know that wealthy aristocrats would have worm mail armour, wielded swords and ridden horses.

Hopefully this helps and one of the resident ancient history flairs can hop in and answer regarding Gallic warfare.