Dominions III is, at heart, a strategic wargame. It is not particularly simple.
Logistics
Their exists a high-level supply model. While the ruler need not act as a quartermaster maintaining depots and allocating resources between baggage trains, he does need to pay attention to keeping his armies well-supplied.
Most troops need to eat; some units (mostly certain magical ones) don't need to eat, but do so anyway. Larger units require more supplies than smaller ones.
Supply is not accumulated or explicitly moved; instead, for each province the game determines the available supply levels. These are based upon such things as the population of the province and the terrain; a well-populated rolling plains provides more foraging possibilties than a sparsely populated mountain pass. In addition, any fortress that belongs to the owner may be able to contribute supplies, if there is a short and uninterrupted path of friendly provinces to the army in question. On the other hand, besieged garrisons must rely on the stores within the fortress, which will surely dwindle over time if the siege is not lifted.
The consequence of providing insufficient supply is starvation. Starvation is not particularly good for armies, and among other things, has a large negative imapct on their morale. In addition, prolonged starvation increases troop mortality through diseases. It is therefore quite unwise for most large armies to penetrate too deeply and risk being surrounded without being able to trace a supply route to a nearby fortress.
It is said that there are mystics who may know ways to alleviate supply shortages. A more mundane if rather short-term approach is to pillage; this can result in seizing food and gold, but quite naturally worsens the attitude of the locals.
Movement
Movement on the strategic scale is measured in provinces. Armies, to remain as cohesive forces, must move at the speed of the slowest unit.
Most infantry can move but one province a month; an army containing only cavalry can usually move two provinces a month, while an army that has only fliers may find itself flying over two provinces to reach a third. A few units may have much faster speeds. Flight is also applicable in battle; in favorable weather conditions, flying units can land behind enemy ranks to slaughter the vulnerable.
Enemy provinces must be secured before an army moves through (not over) it; fortresses will not block movement, but only if besieged. Thus, even a cavalry detachment cannot sweep through multiple enemy-held provinces a turn, while flyers can ignore the intervening provinces to strike behind a front.
Few units can cross the seas at arbitrary points. A few nations have prowess at sailing, and have commanders who can lead fleets (abstracted) with enough supply to cross a small sea to immediately land a quantity of troops on the other side. Other nations have amphibious troops who can walk into the sea... or out of it.
Dominions is not a IGO-UGO game; it is a WEGO game. Movement orders are executed in a not entirely predictable order.