For the purposes of this question, let us assume that you are talking about a retainer 'man-at-arms' up to a landed noble of low standing. I can only answer based on German, French, Spanish, and English nobility.

But yes, he would be highly skilled in a range of weaponry. This would vary from man to man, but at the very least he would be confident in the longsword, the lance, and warhammer. The reason for this is based on the culture of tournament fighting, but also the varying requirements of the medieval battlefield. The pollaxe and warhammer were excellent tools to neutralise armoured opponents, while the longsword was a highly effective weapon against less well protected targets, such as 'serjeants', 'archers', and unarmoured horses. The war lance was an immensely successful shock weapon, able to break through plate armour. Note that a war lance is different from a jousting lance; it more resembles a spear than anything else.

Furthermore, individual nobles might have preferences. Henry V's brother, John of Bedford, famously utilised the pollaxe at Verneuil, in 1424, while Jean Le Maingre expresses his fondness for the war axe in his fitness manuscript.

I feel an important aspect that you should understand is that many of what you would see as 'knights' were semi-professionals, having an income based primarily on land ownership or administration, while others were fundamental career soldiers who were raised to fight and win. Both kinds viewed their weapons as tools of their trade, thereby it is natural that they would become proficient with them. In the same way a carpenter will be taught how to use a saw, a file, a lathe, a knight would learn to use warhammer, axe, sword, and lance.