I can speak to the Roman Army as I produced a couple documentary videos on the subject. (You can find these easily enough with a YouTube search for "Roman Logistics")
TL:DR - All armies lived off the land to some extent to reduce the strain on their logistics or to get things like water and fodder which simply couldn't be easily transported ahead of time. For the Romans, supplies gathered by the army on campaign came from foraging, requisition, and pillaging. Supplies sent to the army came from a logistical system that basically collected goods from across a wide region which were collected at an operational base (typically a port city) which then served as an anchor for the supply line which stretched out to the army with a chain of depots between which convoys could pass. By having more supplies brought to you rather than being gathered on the march you can field larger numbers, you can fight for longer periods, and you can move with more flexibility.
Supplies gathered by the Roman Army
On campaign, the legions obtained these supplies using three means of collection: foraging, requisition, and pillaging.
Foraging involved sending out soldiers to retrieve specific items. Of utmost importance would be sending Aquatores to nearby springs, rivers, lakes, and wells from which they would return to camp with full leather bags, waterskins, barrels, or vessels. Lignatores were sent to gather firewood for cooking, heating, and lighting while Pabulatores collected fodder for the animals. These activities were daily tasks and took many foraging parties. The expeditions of the Frumentatores on the other hand were carried out infrequently. They ventured out in large numbers to collect a wide range of foodstuffs such as wheat, barley, olive oil, wine, and fruit. Roman soldiers might be tasked with reaping grain in the fields or picking figs from orchards.
Requisition involved obtaining supplies from ostensibly friendly forces. These goods were generally brought to the army through seizure, forced purchase, or private markets. A commander might issue orders for inhabitants to supply an army passing through the region. This was often done on a large scale with villages and towns being given specific locations for deposits. Compensation could range from nothing to near market value. More often than not such requisition was a heavy burden on the local population. By contrast, special merchants known as sutlers were far more eager to address the army’s needs. In fact they travelled with the legions and made a living selling them goods. It was through these sutlers that soldiers would get their gourmet items. Garum fish sauce, honey, and spices were always in high demand to improve the taste of rations while items such as fruit, eggs, and fish were great additions to break up the monotony of military rations.
Finally, pillaging involved the seizing of supplies alongside the destruction of property. Such acts typically took place after a battle or siege but might be ordered across enemy territory to punish them or goad them into battle. The Roman military strictly controlled these activities in order to impose discipline and better control the gathering and storage of provisions. In times of lax discipline or civil war however an organized force might quickly become a roving band of robbers. It was said that once a thing got lost under a legionnaire’s cloak there was no power on earth that could snatch it away.
Supplies sent to the Roman Army
You can think of this logistics system as basically a funnel which leads to a pipe that runs all the way to the army. The funnel in this case was the strategic base. This was not a single point but rather the broad source of provisions from outside an area of operations. Generally we are talking about provinces which were assigned specific armies or campaigns to support. These supplies were gathered using various financial means including forced and market purchases, contracting, taxation, and contributions.
This funnel would then be hooked up to the anchor point of the supply pipeline. This was the operational base. As the name implies it supplied the Roman army in its area of operations. It was usually located at a fortified ocean or river port where the largely water-borne supply lines from our funnel could be unloaded on land. These operational bases typically contained infrastructure large enough to hold food and equipment for the campaigning season. If it did not, the Roman army may spend time constructing adequate facilities prior to its campaign. Operational bases were usually permanent for a campaigning season but a new base could be selected if the previous one risked attack or simply if another location proved more convenient.
So now that we have our anchor set up, the next step is to run out that supply pipeline. In a way this would be done by the army itself. What happened was that an army would usually gather at the operational base and just start marching towards its objectives. Along the way, the army would set up its nightly camp. Wherever the army's latest camp was became the tactical base where supplies were sent to and where local supplied were collected. As the army marched on it established a new camp which now became the new tactical base. Its previous camp would often be converted in to a depot with some garrison forces. In this way, as the Roman Army moved it left behind a trail of old bases that got converted into links in our supply line. When supplies came in from the operational base, convoys would then advance along the route and stop routinely at fortified positions where they could rest and resupply. Roads, bridges, and canals were often built to improve the efficiency of these supply routes
Sources
It depends, as there were some armies that had pretty developed baggage trains for example, but in general armies in ancient times relied on pillaging, foraging, trade with merchants that followed the army and more or less coercive supply by allies. Which one they most used highly depends on where they were and who they were. As a typical standard to start off with, I'll describe your average Greek army around the time of the Peloponnesian War:
These armies, centered around the prestigious hoplites^1 could consist of thousands of soldiers, who were generally expected to bring along three days of supplies (at their own expense) at the start of their journey. In addition, many soldiers brought along any servants or animals they could afford. These did not actually go back and forth between friendly territory and the army to form a vein of supplies though, but rather just acted as a mobile supply store that followed closely behind the army itself.
Along with this food, they were expected to bring as much travel money or valuables with them (ἐφόδιον) as possible, to buy food from merchants following the army^2 . These merchants could be seen as an informal, market-driven baggage train, that provided many an army in ancient times with food ^(and wine. always wine), albeit at highly inflated prices, allowing them to sustain an offensive for much longer than would otherwise be possible. When an army found itself in allied territory, it was expected that they be provided by their ally, with or without compensation.
An army in enemy territory could rely on plunder to sustain itself, but this was more of a vulnerability or extra, and often required good timing, by either invading enemy land at the exact time the crops are ripe and stealing their crops, or by successfully surprising enemy villages, before they had a chance of emptying their food stores. Both methods of plunder were unreliable, and even successful plundering could only alleviate the food requirements of an army for only a short while. Failing to acquire plunder could spell the near-immediate end of a campaign.^3
Having soldiers grow food during a long siege is out of the question; most wars were deliberately fought during periods were farmers didn't have much to do on the land, and kept short by necessity to allow the working classes to return to their fields and crafts and businesses, which they could only abandon for so long. If a war or siege lasted through winter, the army was typically reduced to only a fraction of its nominal size, indicating how hard (or perhaps unprofitable/damaging) it was to sustain any soldiers during those months.
Now of course there are exceptions, and although I can't cover all of them, there are some very notable ones. The Spartan armies were unique among the Greeks for having something resembling army professionalism, and this is, among other things, reflected in the more (state) organized baggage trains, which were actually protected by helot hoplites. The Romans are apparently an exception to this as well, though their supply trains were still based on immediate requirements rather than organized tables with the expected requirements like we see later on. I could add more about the Roman supply system but it's probably out of my league and someone else might do a much better job.
For Greek armies typically only the hoplite numbers are noted in sources, but the uncounted lighter infantry (peltasts and such) would make up more than half of the actual army size. Because this light infantry was so much less prestigious (meaning less is written about them) and presumably poorer as well, it's not as clear how they were provided with the necessary supplies.
Mentioned (among other places) in Education of Cyrus, book 6
Thucydides 4.6 : the Spartans run out of food in less than 15 days because the crops of the territory they invade are not edible yet.