So, this is a question that has been asked since literally the second day of Joan's (first) trial, February 22. The tl;dr is that Joan denies it repeatedly over the course of the questioning, but it still ends up being one of the formal articles of accusation when they are quite belatedly issued at the end of March.

Joan was a fairly well-off peasant from Domremy, a village in an area loyal to France. Did she learn swordfighting as a child?

Asked whether she had learned any skill in her youth, she said yes, to sew linen and to spin; and she feared no woman in Rouen for sewing and spinning. (Feb 22)

When she had to flee her home village on account of Burgundian raids, she ended up in Neufchateau, where a major point of contention between Joan and her accusers comes up. Some days, she says, she was responsible for taking sheep and cows out to pasture in the meadow. (Feb 24) The judges were quite sure she learned to wield weapons out there:

Article 8 (March 27, articles of accusation read) ...While staying at this inn, Joan would sometimes keep to the women, but sometimes she would take the sheep to the field and lead the horses to water, and to meadows and pastures. And there she learned to ride and gained knowledge of arms. With regard to this article, Joan says she refers to her previous answers. She denies the rest.

It is, of course, heavily possible that somewhere along the way of Joan's time at the French court or with the army, someone would at least have offered fighting or other lessons to her. But Joan testified that she carried no sword in battle:

Asked whether, when she went to Orleans, she had a banner...she said she had a banner. [describes a banner filled with religious imagery] Asked which she preferred, her banner or sword, she said she was much fonder, indeed forty times fonder, of the banner than of the sword...She carried the banner when she attacked the enemies, to avoid killing anyone; she says she never did kill anyone. (Feb 27)

Now, about the iconography--as you say, Joan is frequently depicted with a sword. (To say nothing of the bit about the armor shirt with a full length skirt. Note she's wearing a sword, but holding the banner.) Did Joan have a sword? Yes, and the answer is so very late medieval:

She says she had a sword that she took at Vaucouleurs. When she was at Tours or at Chinon, she added, she sent to find a sword in the church of Sainte-Catherine de Fierbois, behind the altar; and it was immediately found, completely rusted. Asked how she knew the sword was there, she said the sword was in the ground, rusted, bearing five crosses; she knew from the voices that it was there...She wrote to the clergy of the place asking them to please let her have the sword, which they sent her...She says as soon as the sword was found, the clergy there rubbed it and at once the rust fell off effortlessly. [...]

Asked whether she had her sword with her when she was captured, she said no, but she had a sword that was taken from a Burgundian. (Feb 27)

She refers several times to receiving gifts of arms and horses from the king as well.

So: if we are willing to believe the words of a 19-year-old woman absolutely convicted of her holy calling and on trial for her life, no, Joan had no military training. On the battlefield, she insisted on being present for the decision-making, but there were trained strategists calling the shots. She carried weapons, but ultimately relied on her power as God's vessel to inspire the soldiers to fight for France.

Sources: All quotations are from Dan Hobbins' translation The Trial of Joan of Arc*, which is based on the Latin text written immediately after the trial as a translation of the French "stenographer's" record (which only survives in later copies--Hobbins compared the Latin to the later French and notes the few points of divergence). It's an incredibly readable translation. And even though Hobbins' interest (as laid out in the intro) is the trial itself as medieval legal procedure, you still get quite an impression of Joan herself through the answers.*