Dominions III is, at heart, a strategic wargame. It is not particularly simple.
Dominions implements 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 on the strategic map is measured in provinces. Armies move at the speed of the slowest unit.
Most heavy infantry can move but one province a month, while light infantry generally can move up to two. An army containing only cavalry can usually move 2-3 provinces a month. Some magical spells can enable units or armies to travel even much further distances in one turn.
Some units have natural flying ability, while others can gain such through magical means (Winged Shoes, Stymphalian Wings, Magic Carpet). An army only comprised of flying units can fly over up two intervening friendly or enemy provinces to reach a third. The intervening provinces must be land provinces, but the destination can be land or sea. Flying troops always "land" at their destination, even if that province is a sea. In such a case, beware that your army has the means to breathe underwater!
Fliers also have the advantage that difficult terrain such as forests and mountains does not slow them down, even if they lack familiarity with such regions. Flight is also applicable in battle; in favorable weather conditions, flying units can land behind enemy ranks to slaughter vulnerable magi.
A few nations (ie. Vanheim) have prowess at sailing, possessing commanders that enable a land army to cross a sea province and immediately land their troops on the other side. Their is also a forgable magical item that allows the same (the Pocket Ship, Const-6 N3A2). Other nations have amphibious troops who can walk into the sea... or out of it. Some magically forged items allow land-based commanders or even entire armies to breath underwater, and vice-versa. Finally there are some rare creatures and magi who allow those accompanying them access to the sea. Underwater movement is slow -- only one province per turn is permitted.
Stealthy commanders leading either only stealthy troops or none at all can attempt to move to an enemy-controlled province without triggering a fight. It should be noted that the province militia and any patrollers have a chance of detecting such forces, and this chance is per commander; thus, stealthy forces may be destroyed piecemeal when detected, as their compatriots will not join in.
Dominions is not a IGO-UGO game; it is a WEGO game. Movement orders are executed in a not entirely predictable order.