While Urban centres contain the industrial might and high concentrations of population which are vital for winning a civil war, they do all have to be fed. How did the Reds ensure they got regular food shipments from the outlying farms, and were partisans and resistance a problem in getting food sent to Leningrad or Moscow?
More generally: How do resource flows work in modern-era civil wars anyways? Does the rebelling side usually manufacture their own arms or at least try to, and how do they get the resources to do so? How do you get food to the population if your country is a food importer?
You've identified a key problem that the Bolsheviks faced during the Civil War. The Russian economy had already been devastated during the Great War, and much of the country already faced famine by 1917 (this being a large reason for the revolutions in the first place). And then, as you said, when the Civil War began in earnest towards 1918, the Reds had the problem of feeding the major urban centers in European Russia.
The Bolsheviks resolved their problem by implementing "War Communism". It was essentially a set of emergency measures designed first and foremost to feed the Red Army and the cities by nationalizing production and agriculture. The state centralized all work and and took over as sole producer and distributor.
Because of the chaotic situation, private ownership was not totally wiped out, but the Bolsheviks did succeed largely in requisitioning enough food to feed their armies and the cities. In practice, this was done simply by confiscating whatever they needed from the countryside. It left thousands of peasants starving to death. Again, as you guessed, partisans were a major problem. In fact, several major uprisings occurred against Bolshevik requisitioning, most notably in Tambov where a rebel Blue Army some 40,000 strong fought against the Reds.
On the topic of where the Reds got their arms and supplies, they had the luck of controlling the vast majority of Tsarist military supplies. The Whites had to scramble for foreign aid while the Bolsheviks could mostly rely on Great War stocks.
TL/DR The Bolsheviks fed their cities and troops by shamelessly taking food from the countryside, and ruthlessly crushing any opposition to it.
Source: Richard Pipes, Russia Under the Bolshevik Regime Also, you might enjoy a look at Solzhenitsyn's book of short stories, Apricot Jam. One of the stories, called Ego, is about a Tambov peasant who rebels against Bolshevik requisitioning.
I'd like to add a little bit to u/CaptainPyjamaShark's excellent information. War Communism, for the peasants, meant a high tax upon their produce. While their were specific figures for how much was to be taken out of a crop yield, in practice, it was far more complicated. Farmers would resort to all sorts of methods to hide their produce from Cheka agents (Soviet Secret Police) and Red Army soldiers. This worked for a little while, but then the government started looking at how much land was tilled. By the end of the Civil War, peasants just stopped producing food beyond what they needed. In their eyes, why work hard to have the fruits of your labor taken from you? In result, as CPS noted, forced requisition quickly became whatever the troops or agents could find. Finally, by the end of the Civil War (the main bit in 1921). The farmers were fed up with the Bolsheviks. Rebellion broke out in Petrograd and at the Kronstadt Naval Base. The Petrograd rebellion was quickly crushed, and due to a host of errors, the Naval Base was overwhelmed by Red Army troops. Facing threat of yet another civil war, the Bolshevik government declared an end to War Communism and a simple levy tax on farmers. This, for the vast majority, was exactly what the farmers wanted; and thoughts of rebellion ended.