Disclaimer: I am not a historian, just a history enthusiast. The following response comes from my school history lessons and some personal research. As the history lessons were state sponsored, there is a good chance of them being biased. I have verified some of the facts from Hitti's "History of the Arabs"

Pre (and for that matter, post and during) Islamic Arabs were tribal in nature and feuds over perceived tribal honor violations were relatively common. Thus every Arab received some sort of martial training, most likely with a sword. The battle of Uhad mentions the use of archers and spoils in the form of armor so the use of those may also have been trained. Those who owned horses would have been trained in their use as well (I don't know what weapons or tactics though)

I'll restrict my response to the first 3 major battles fought during Mohammad's time between the Muslims (who had largely migrated to Medina) and the Quriash tribe and allies of Mekkah. These were the battles of Badr, Uhad and the so called Battle of the Trench. These conflicts were very personal in nature since the people from both sides knew each other. In a lot of cases, people from the same family were on opposite sides (e.g. Muhammad was of the Quraish tribe). Armies were led by tribal leaders who showed particular organizational or martial talent.

Initially both armies would face each other. One or more challengers would walk out to the center and the opposite army would send out their own champions. These would square of one on one undisturbed in a fight to the death. This would repeat until both sides joined in a general melee (this last is a speculation on my part). The Battle of Badr was fought in the manner and resulted in a 300 strong Muslim army defeating a 1000 strong Quraish army.

The Battle of Uhad was marked by the tactical use of archers. Again I'm only quoting what I learned so I don't know their actual significance. In any case, the story goes that before the start of the battle, Muhammad stationed some archers on the high ground to prevent his army being flanked. The battle took place in a similar manner to Badr (700 Muslims on 3000 Quraish this time.) and the Muslims looked to win. The Quraish were losing so badly that they fled the field, stripping themselves of their armor while the Muslims gave chase. Seeing the spoils left behind, the archers abandoned their position to collect, leaving the Muslims vulnerable to a flank attack which is exactly what happened. It ended in a stalemate lopsided towards the Quraish.

The Battle of the Trench was the seige of Medina by a comparatively large force of Quraish. To aid in its defence the Muslims dug a trench around Medina. The seige was unsuccessful and Mekkans finally broke away after disease affected their ranks. Despite this being a siege, there are, I believe, still reports of the one-on-one challenges.