We don't have, so far as I know, any first hand account of battle from a common footsoldier from the medieval period, so we don't have their perspective on whether or not they were frightened. We can, however, be pretty sure that they were afraid. Why? Well, apart from basic human psychology, we do have a few examples of both the nobility being afraid and of chroniclers who witnessed battles first hand noting fear.

For example, Geoffroi de Charny, foremost knight in France during the early and mid 14th century, wrote very specifically about the fear felt in battle in his Book of Charny, a work intended for prospective knights:

In this profession one has to endure heat, hunger and hard work, to sleep little and often to keep watch. And to be exhausted and to sleep uncomfortably on the ground only to be abruptly awakened. And you will be powerless to change the situation. You will often be afraid when you see your enemies coming towards you with lowered lances to run you through and with drawn swords to cut you down. Bolts and arrows come at you and you do not know how best to protect yourself. You see people killing each other, fleeing, dying and being taken prisoner and you see the bodies of your dead friends lying before you. But your horse is not dead, and by its vigorous speed you can escape in dishonour. But if you stay, you will win eternal honour. Is he not a great martyr, who puts himself to such work?

The Itinerarium Regis Ricardi records that, although the crusader foot did constantly repel Saracen charges during the Battle of Arsuf, many of the crossbowmen did abandon the battle lines out of fear:

Oh! how useful to us on that day were our arbalesters and bowmen, who closed the extremities of the lines, and did their best to repel the obstinate Turks. The enemy came rushing down, like a torrent, to the attack; and many of our arbalesters, unable to sustain the weight of their terrible and calamitous charge, threw away their arms, and fearing lest they should be shut out, took refuge, In crowds, behind the dense lines of the army; yielding, through fear of death, to sufferings which they could not support.

While I don't have a translation of the appropriate chronicle to quote from, something similar happened at the Battle of the Standard: the English knights were mixed in with the English infantry and resisted the Scottish attacks, but the ordinary infantry were pushed back two or three times and had to be rallied each time (The Great Warbow).

Likewise, the Rule of the Knights Templar indicates that serjeants (footsoldiers) who did not have mail armour were permitted to flee from battle, but that those who did wear it were expected to stay so long as any Christian banner was on the field, just as the knights were expected to do (The Art of Warfare in Western Europe). Taken with various other literary evidence, J.F. Verbruggen takes this to indicate that armour was a significant boost to morale and countered fear to a degree.

As to the arms and equipment of a common soldier, they very much depended on their circumstances, but rarely would just be just his father's sword. Anywhere up to half the fighting population of a 14th century town would have had full or partial armour, along with weapons that likely included spear and shield or a polearm, while professional infantry (mercenary or stipendiary) seem to have had mail from the mid-11th century on (The Deeds of William, duke of the Normans and the King of the English). Of course, levied rural militias still had a lot of men without good armour, as seen by Edward II's attempts to increase the proportion of armour men being raised via Comissions of Array, but the fighting elements of medieval armies were largely made up of professional or semi-professional soldiers and the better armed levies in any case.

There has already been a fantastic answer to this question but I thought, seeing as I could, that I would make an addition of my own. It is important to understand that Medieval Warfare was not fixed, it evolved drastically and rapidly, particularly in the Later Middle Ages with what Clifford Rogers has labelled the Later Medieval 'Military Revolution'. To be more precise he is talking about the change in warfare which occurred during The Hundred Years War, where previously cavalry in the form of the knightly classes dominated the battlefield now archers and crucially close masses of men armed with pikes lined up to receive the incoming charge. The first proper example of this is at the Battle of Courtrai in Flanders in 1302 when the Flower of French chivalry ran headlong into the massed pike formation. This led to the death of 1,000 of the 2,500 knights who had charged. Rogers labelled this the ‘first clang of the death knoll of heavy cavalry as the dominant force on the battlefields of western Europe’.
This is important as it fundamentally challenged the way warfare operated, especially when it came to the taking of prisoners, which became increasingly more difficult as the lessons learned at Courtrai began to be implemented. While perhaps not the concern of the common footsoldier, (who would want too take them prisoner)except in the case where they could capture a well known knight or Lord, the inability to take prisoners due to the tight crush of bodies as two great hedgehogs clashed on the field while arrows reined down around them made conflict far more bloody and contributed to the end of 'chivalry'.
An example of when the melee, was so awful and horrible for the common footsoldier was at Dupplin Moor in 1332 when the new massed infantry formations led to a huge crush which led to men being suffocated or crushed to death. Also, Geoffrey le Baker describes how at Crécy, in 1346 ‘in the middle of the French army, many were crushed to death without any wound, smothered by the weight of the crowd.’ In short certainly in the Later Middle Ages combat for the common footsoldier was undoubtedly terrifying, being hemmed in, surrounded by ranks and ranks of men and having nowhere to run as the crush set in and the arrows blackened the sky. As much as I feel I probably shouldn't use this example, if you have seen the Battle of the Bastards episode of Game of Thrones, then you will at least have a visual of how appalling it would have been. Juxtapose this with the Flanders War of 1127 and, where 1,000 knights fought and saw only one casualty and you can also see how much warfare had changed.