The earliest recorded battle is one between the Sumerian city states of Lagash and Umma from around 2450 BCE that was depicted in the so called Vulture Stele, named due to the image of vultures feasting on the dead of Umma. As you can see from the pictures on the museum site, the soldiers of Lagash are depicted with broad shields and in a tight, phalanx like formation. Of course it is a bit of a leap to go from this to saying these are professional soldiers, but at least we can say they were organized as an army.

This was not, of course, the first war, or even the first war we know about. That honor probably goes to the unification of Egypt by king Narmer, which is depicted in the famous Narmer Palette--although "depicted" may be too strong of a word, unless Early Egyptian warfare involved far more leopard-snake-giraffe hybrids than I have been lead to believe. The Sumerian kings list mentions one Atab "consolidating foreign lands" and Enmebaragesi "subduing Elam" but, well, if you just click that link you can see why this might not be the most straightforward source. And there are also archaeological remains that look a lot like battles, but of course we can't really understand the political context of this.

One of the most interesting examples of a professional army comes from the funerary inscription of Ahmose son of Ebana, a military officer who conveniently/confusingly served under the Pharaoh Ahmose I, who expelled the Hyksos from Egypt and ushered the the New Kingdom. he was a soldier and son of a soldier who "followed the sovereign on foot when he rode about on his chariot" but rose through the ranks and ended up a fairly wealthy and comfortably situated man due to the rewards that pharaoh gave him. In fact, despite Egypt's peaceful reputation many of the earliest good descriptions of warfare come from there, such as the description of the Battle of Megiddo by the pharaoh Thutmose III. This is, I believe, the first highly detailed description of a battle, where we can talk about formations and make strategic assessments.

Before those last two Egyptian battles it is nearly impossible to give an even halfway decent account of war, as battles are described in allusive, highly mythologized terms. The Victory Stele of Naram-Sin, for example, shows the king as a literally deified person, towering over others and with horns on his head. It isn't until much later that reliable battle narratives become the norm, and so actually assessing what war looked like is impossibly difficult. Not that this stops anyone.