How was such a great army, like the army of Napoleon on its way to Moscow, supplied with food and other needed things? Were they transporting it from France or maybe stealing from territory on their way? What were general methods to supply great armies?
Campaigns of Napoleon: The Mind and Method of History's Greatest Soldier by David Chandler is a good source of informations on Napoleon. See pages 749-797, chapter "War Pans and Preparations".
Mostly, the supplies were stored and ran from Poland. However, the army was accompanied by no less than 200,000 animals and 250,000 vehicles. A lot of the supplies were to be found on site -- hence the delay in the attack. The sheer scale of Russia and its scorched earth were massive factors in the failure of the offensive.
Napoleon's army got some supplies from occupied and allied territories of course, especially from Prussia and Poland. Still, the distances were too long to get sufficient supplies in and so the soldiers plundered villages on their way to collect food. This turned out particularly devastating on their way back: the Russians used scorched earth tactics and the army literally starved to death. To quote Wikipedia:
There were four main methods of supplying troops during the Napoleonic period;
(1) The individual soldiers would be issued with rations that they carried in their knapsack. Enough for about two weeks. This was often in the form of biscuits.
(2) Cattle or other animals were herded along and slaughtered along the way but these increased the grazing requirement.
(3) Wagons following in the supply train. The main component of which was normally flour. Periodically they would stop construct ovens and bake biscuits. (In Spain, Wellington's supply was mainly on mules rather than wagons)
(4) Living off the land and foraging. French armies tended to rely on this much more than the various allied armies. In western Europe this was much more successful than in less rich regions like Spain or Russia. While foraging can often enable armies to move faster, the prolonged use of foraging brings down good army discipline as troops get very used to this foraging and just outright looting follows and result in bad relations with civilians.(As well as opportunities for the less committed soldiers to just desert)
For the 1812 invasion of Russia, it was gathered in eastern regions under Napoleon's control and sent to the areas the Army was concentrating in. This was by wagon mainly but there was some use of river transport. Napoleon never intended to march to Moscow. The campaign was a series of ad-hoc decisions by Napoleon to move further into Russia which he had not planned for and had not the supply to do. Napoleon only had the wagon capacity to supply around half his army that sort of distance, and that's ideally. There were problems right from the start. Horses were not in good condition when they started (and the French cavalry were notorious for their bad treatment of their horses), and the extravagant use and marching of Murat was not going to conserve horses.
On the retreat from Moscow, the shortage of horses was made worse by poor march discipline, with much looted baggage that consumed both space on the road (making for more congestion slower march) and use of horses. Napoleon had been urged by his Generals to abandon much of the artillery (particularly those of the less useful smaller calibres) to free up some horses but this was rejected.