We tried to design Hegemony to be fairly intuitive for players who understand the logistics of military campaigning.
Make sure your army is well supplied. Load a worker unit with food and have it follow your army, screening with scout cavalry.
-The worker doesn't use food itself.
-The combat units will automatically feed off the worker units food and pass it around to other combat units.
-The scout cavalry can dash ahead and grab flocks of sheep and other vulnerable points.
-Bring along a siege train of some catapults to shorten city siege times. (the longer the siege the more food you need)
-When you capture an enemy city, move in well supplied units to fill the city with food.
-The city needs food for the garrison to have any effect on preventing rebellion. (the key point to watch)
When workers are loaded with food and tag along with an army, they are acting like camp-followers. For extended campaigns into the sparsely farmed northern regions, or if I want/need to start a campaign in winter, I have started with as many as one fully loaded worker for every two heavy infantry units.
March into enemy areas around harvest time to keep well supplied.
You can also have flocks of sheep follow your armies. Combat units will automatically feed off them, the same way they feed off food-laden workers, except that the sheep permanently disappear.