We don't know definitively what Celtic culture was like because there was no "definitive" Celtic culture. At best, the groups we label "Celtic" shared linguistic ties and certain loose cultural features. We can't really generalize further than that because these groups were multi-ethnic, adapted certain features from other cultures, and innovated themselves over time. The very lifestyle of these peoples varied from Poland to Ireland, and from pre-history to the early Medieval period. The term Celt was only ever maybe used by a few tribes, and terms like Κελτοί (Keltoi) and Galli were applied to these tribes by Greeks and Romans, respectively. While coalitions of these tribes may have eventually begun to unify under common concepts of identity or at least familiarity by the Classical period, this certainly did not encompass the broad range of peoples labelled "Celtic" today. It's commonly noted that inhabitants of Britain and Ireland weren't considered Celts like the people of Continental Europe were in Antiquity, it was only from the 15th Century AD onwards that antiquarians started calling ancient British and Irish tribes Celtic.

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I can still answer your question about ancient Celtic armaments, but I wanted to break that common misconception first. Firstly, it is true that the Celts probably never went into battle naked and that the Romans played up the idea of the savage warrior people. The concept of a barbarous horde of warriors who were madly courageous in victory, and sickeningly cowardly in defeat, was great propaganda. It not only explained why the Romans were so valiant for conquering them, it even helps illustrate why the Celts "deserved" to be conquered.

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Most depictions of Celtic warriors come from the Hellenistic period or the following Roman period. There just aren't that many surviving examples of Celtic art compared to Greek or Roman finds. The Greeks and Romans also generally didn't even care much about the existence of the Celts until around the 4th and 3rd Centuries BC, when massive coalitions of Celtic tribes migrated southeast and attempted (mostly unsuccessfully) to settle in parts of southeastern Europe and western Asia.

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Many depictions of Celts show them wearing mail armour and carrying oval shields which were long enough to protect the greater part of the body. In addition to chainmail, Celts wore woven breeches, tunics, and cloaks which were usually made with wool or animal hair. Surviving scraps of Celtic textiles reveal that these were brightly dyed with plant dyes, and patterns such as plaid and tweed were quite popular. The torc, an iconic gold necklace, is also an identifying feature of high status Celts, and ornate brooches were used to fasten clothing.

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The 1st Century BC Roman author Varro claimed that the Celts invented mail armour, and areas with Celtic presences like northern Italy are likely where Romans were first introduced to mail. Only a few Celtic mail shirts survive, but these usually fall below the waist and have broad shoulder straps. Leather cuirasses actually are attested to by artistic evidence, as are cloth tunics. Leather armour may not have been common, but it certainly existed.

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This 2nd Century AD terracotta statue depicts an archetypal Celtic warrior. He has long hair, a drooping moustache, a long oval shield, and appears to be wearing some kind of (possibly chainmail) tunic. Images of Celts in breastplates also exist however. One of the best pre-Roman depictions of a Gallic (Celtic) warrior is Le Guerrier de Grezan. As you can see from the image I linked on Flickr, he is wearing a breastplate, a torc necklace, a sun pendant, and a kind of hood or helmet.

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It's not exactly what the "common man" could have afforded. When historians speak of Celtic warriors they usually focus on a member of a (possibly) warrior-aristocracy. These warriors are known from the richness of their burials, and the testimony of Greek/Roman authors. They could afford to invest in expensive equipment, since warfare and raids would have been the way that they maintained their status and even moved up in status. Although it's easy to forget that these burials are exceptional and these accounts are embellished, it is necessary to remember that the warriors described are uncommon.

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La Tene burials of warriors often include swords, spears, knives, drinking horns, dishes, and brooches. More important burials can include chariots or horses. According to Romans, the Celts used these chariots to get into the fray and intimidate their enemies before jumping off and fighting on foot. Horses were status symbols, and they were also integral to Celtic combat. According to Pausanias, there was a custom where chieftains would ride into combat with three horses, accompanied by two grooms. If the chieftain's horse died, one of the grooms would give him another, and if he died, one of the grooms mounted and fought in his place. There are some parallels to the use of chariots and horses in other areas of Europe like Iron Age Greece, but it might be wise not to dive too far into reconstructive comparisons.

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Swords, rather than spears, are more commonly represented but spears were also very common weapons. Studies of weapons found in Gaul indicate that there was a consistent style of arms in production between the 5th and 1st Century BC. This means that from the Late Iron Age to the Roman period, Celts were making and using arms that were consistent in approximate shape and size. Celtic swords were usually longer and more well accustomed to cutting than stabbing, making them different from the swords common to Greece or southern Italy. Spears with shafts of ash are common, and an impressive degree of variation and innovation in spearhead design is attested to. Changes in design varied between regions, with some being quite localized. For example, British shields and helmets are often more ornate than those of the Continent.

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Swords survive better than spears or shields because iron does not decompose as fast as wood or leather (duh) but this means that we can't actually say how common certain weapons were relative to others based solely on archaeological deposits of weapons. After all, spears and shields were essential based on literary and artistic evidence, and it only makes sense that these arms would have been useful in the kind of warfare Celts were accustomed to.

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A kind of hierarchy of warriors appears on the Gundestrup Cauldron from Jutland, Denmark.A man with a boar-crested helmet and a sword follows the spearmen, and behind him are three carnyx players. At the far left is an oversized god dipping a man in a cauldron of rebirth. In the top register, a bunch of warriors or chieftains on horseback ride away from the god. It seems like this scene portrays warriors moving up in status in the afterlife, they are reborn and apparently promoted at the same time.

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All of the men appear to be wearing tunics and short breeches, but some differences in equipment exist. The mounted warriors and the man leading the carnyx players appear to have better or more ornate helmets for example, and horses would have meant they were of higher status. Helmets are well attested in La Tene and Halstatt burials, and were certainly used but they are not as common as one might assume. This ties into Greek and Roman claims that many Celtic peoples avoided the use of helmets for cultural reasons. Bronze helmets like this Negova Type or the later Meyrick Helmet exist, and many are more ornate like the clearly ceremonial Agris Helmt. The use of materials like gold and coral in addition to iron and bronze makes it clear that many of these ceremonial helmets were status symbols, and their often impractical designs indicate that they may have been crafted with an eye towards visibility (perhaps during parades) as opposed to protection of the head in combat. Later La Tene helmets actually developed to be less ornate and more practical based on studies of helmets found in La Tene burials, although Classical writers preferred to talk about the more visually striking helmets.