'Barbarians' were generally quite experienced in warfare. Fighting in formation makes sense. Do you expect something else?

Let's take the inhabitants of Denmark during the 1-4th centuries as our example (they are quite typical 'barbarians', I'd say). We have relatively good archaeological information on them, most importantly because they had a practice where after the defeat of an enemy army, they sacrificed the entire army equipment in a bog for us archaeologists to find. There are a dozen or so of these massive weapon sacrifices in Southern Scandinavia, from all throughout the Roman Iron Age. The one at Illerup Ådal is the biggest (because it's been excavated more extensively than many of the others), and it dates from the early part of the 2nd century; basically a time when the Empire was at its height, in the middle of the Pax Romana. This already shows that this 'pax' only referred to the Empire itself, and barbarians outside of it were still going at it.

Now, from this army equipment emerges quite a different picture from what the Roman 'ethnographers' (primarily Tacitus) write. Instead of a disorganised horde, it looks like warriors actually had equipment quite similar to the Romans. In fact, the normal soldier's outfit was standardized as one stabbing lance, one throwing javelin, a shield, and a short knife, with a military belt with a utility knife and a firestarter kit. However, we also see that besides this normal army outfit, there was also a smaller group of warriors that had this same equipment, but with bronze decoration (particularly a bronze shield-boss). There is an even smaller number of shields that have silver decoration, and silver decorations on the scabbard and military belt. There were very few items that were decorated with gold.

From this, and the ratios between the different materials, we can deduce that there was such a thing as organised ranks within the army. It seems that there was a chain of command, expressed in the equipment, from an army leader, to his lieutenants, to the sergeants, to the common soldier.

Yet more precise analysis shows that there were also some special groups within the army. There is a relatively small number of soldiers that also has a sword, some archers, and some horse-riders. The last category is so small that it is probably just the commanders who were riding, but the category of archers is a special group within the army.

Now, analysis of the Ejsbøl and Nydam weapon sacrifices gives even more detail. It seems that rather than a Roman formal system of hierarchy, the barbarian system was organised on a principle of 'königsnähe', or 'proximity to the king'. Those who are physically and in rank closer to the single commander of the entire army carry higher-ranked equipment, and they control a piramidal hierarchy below themselves. Basically, status radiates out from the general. This is different from the Roman army, where a status is institutionalised and independent of the commander; a centurion is still a centurion, regardless of whether he serves under Caesar or Pompey.

If we then take a step back and look at the actual equipment, it seems that everything was highly standardized. Swords and spears all look the same. However, unlike the Roman army, this barbarian army had their weapons 'customised'. While sword blades were all the same, there was a great variety in shapes and decorations on sword-hilts, for example. Similarly, spears had their own decorations, and sometimes personal names scratched in. This suggests that while the equipment was supplied centrally, it was still owned privately.

Funny thing is that most of the weapons were actually made in Roman factories, so this army was mostly equipped by Roman arms trade. A bit similar to how an army in Liberia or Somalia today fights with primarily Russian or American weapons, sent there to project influence (and perhaps stabilize distant opponents) by the superpowers. This is what Romans called 'divide and conquer' policy.

Note also that shield bosses were round, and did not have a shield spike like earlier Celtic and later Migration period shields had. It has been suggested (but I'm not sure if I buy that) that the spike was disused because if you are fighting in close formation, you might accidentally stab the person in front of you with it. So the existence of predominantly round shield bosses might suggest that the shields were adapted for formation fighting.

If we compare this image from the archaeology with actual depictions of barbarian armies, made by the barbarians themselves (so not caricatures made by the Romans), it actually correspondents rather well. The Gundestrup kettle has the best depiction of a group of warriors in formation, and on these, you see the same standardized equipment.

So yea, it is most likely that the barbarians did, in fact, know how to fight in formation. Whether they needed Romans to teach them this I should leave up to your own imagination.

Depends on the barbarians. Most of the late antique barbarians had actually served in the Roman army as allied troops at one time or another, so they had plenty of time to learn techniques from the Romans if they hadn't been using them before.

If you want to learn more about Attila and his time you can check out Christopher Kelly's The End of Empire.

While the Carolingian era is after the Franks were no longer being described as barbarians, the Franks of that era used many of the same tactics as the Romans, following Vegetius as their chief guide for how to run an army. Their cavalry are noted as having had the ability to turn around en masse at a gallop while maintaining formation.

Source: Bernard Bachrach, Early Carolingian Warfare.