Did the soldiers use to travel light, with their weapon and equipment packed in carts?

Normally yes. Unless they were expecting to fight, or in dangerous areas, they would walk in normal clothes, or possibly with a gambeson (paded jacket that formed the base for most forms of armour) if they wanted the warmth and/or some protection should they be attacked, but not wanting to carry their full armour.

Armour is actualy easier to wear than carry if you have to transport it yourself, the weight is much better distributed whilst wearing it than on your shoulders. A chain hauberk is belted at the waist to take some of the weight there, and plate armour tapers in to the natural waist too, so the weight isnt all on your shoulders. It's less strain when its on your waist, which is why rucksacks sometimes have hip straps for hiking. Plate also weighs less than the rucksacks soldiers in Afghanistan are wearing. Here's a video showing the weight of early 15th century armour, and a comparison by the same guy to modern equipment, and he describes at ~13 minutes a comparison to a study done by the USMC on carrying weight on the march.

 

As for whether weapons were carried with them, that varies based on what time period, which army, and where they were, so the safest answer is sometimes weapons were in carts, sometimes carried. Or the main weapon (such as a polearm for a man-at-arms) would be in the cart and the man-at-arms would wear a sidearm if they had one, like a sword or rondel dagger. It's worth noting that knights or wealthier man-at-arms would have a retinue that would follow them, either fighting with them in battle or simply to prepare the camp and maintain the knights weapons and feed him.

But an army would almost always have a large baggage train behind it, with camp followers who would set up the camps/prepare food etc. Again, this varies on time and which army. One example of a previous question about them on this sub.