For many armies, it was all about foraging and living off the 'land'. Unfortunately for many non-soldiers in the area, this often included their food stores, animals, harvests, etc.

In some places and time periods, if the soldiers were more professional and the army as a whole more disciplined, the pillaging would be toned down to simply requisitioning supplies from towns and villages. Maybe payment would even be given in exchange.

But probably not.

And of course, supply trains.

In Ancient China, supply lines were a constant source of worry as well as battle, because strategists before and since Sun Tzu knew well from history and experience that starving an army was one of the most effective ways to win a war.

As an army advanced, supply relay stations would have been built if they were not already existing. In settled lands, they might have occupied a village or town and use it for their supply routes.

In rougher terrain they would often erect a simple palisade as well as build simple grain stores, bury edible roots and tubers, hedging in animals, etc.

Soldiers would be left behind to guard these valuable supplies, patrols set up, and a quarter master would remain to manage the logistics.

But this was actually not the most ideal supply route for the Chinese. The best way would be to travel by water. The many rivers in China are famous not just because of the fertile floods they brought but also because they were awesome avenues of trade - and warfare.

Armies would often advance along or at least stay near the rivers if they were not already riding down the rivers on boats to speed their advance.

This is why navies were so important for various Chinese kingdoms. The lords who controlled the major water lanes of China had the easiest way to transport soldiers, equipment and supplies as well as a great source of lucrative trade and lucrative taxes.

Travel by water was much faster and often safer than travel by land, especially if they feared ambushes by bandits or enemy raiders. If one could maintain their grip on the river, they had an almost assured flow of steady supplies and reinforcements.

The coastline was also important but of course, harder to control and defend. Also, pirates were always a serious concern. Storms were also more worrisome out at sea than on the more predictable rivers, even if they stuck close to the shoreline.

As for the food itself, rice rations were easy to transport. They could be stored for long periods when dried and/or treated and prepared on site with some water. Various other grains would supplement the diet.

Various roots and tubers were also easy to store and carry along for campaigns.

Vegetables were harder to keep fresh but pickling helped a lot.

Meat would have been harder but in general they would have just brought live animals along for the trip and slaughter them as they needed.

Beans were also hugely important, and another easy to store and transport since they would not spoil quickly when kept dry.

Clean water was a serious issue though and sickness because of the lack thereof was a serious problem throughout much of military history anywhere. This made traveling near the river, where the water was potable, even more important.

The simplest means of food storage and preparation was grains, and the most effective and nearly universal form of storing those grains was a type of bread or biscuit called 'hardtack'. This is basically a piece of bread so baked and dry that it can last literally for years. It was wholey terrible if you just tried to eat it raw, but added calories to things like broths and could be dipped in water, beer, whatever, to make it edible. Hardtack like breads have been a standard element of armies on the move since at least the Roman times, and likely farther back.

That said, for an army on the march in a hostile territory, a large portion of things like fresh meat would be appropriated by the soldiers from the local population. The more mobile an army was required to be, the less ability it had to rely on supply trains, and this is something that has happened as recently as the Battle of the Bulge in 1944, where German planning relied on stealing supplies from the Allies. However, a good example from Medieval times would be Edward's raids into France in the prelude to the Battle of Poitiers in 1356, where his armies lived off the land.

The reality of medieval warfare was that very few leaders were that concerned with the logistics issues, and indeed, they were the exception rather than the rule. Supplies like food were often left to the men themselves, who often were organized into groups with parts of that group always out foraging, hunting, and stealing. This is, of course, one reason medieval warfare was so hard on both the troops as well as the populace.

If a commander did take steps to supply their army, it was most often in the form of a naval presence rather than a large supply train. Ships following an army along a coast or a river have long been the primary form of resupplying armies deep in the field.