Fire Lord Sozin wanted world domination, not the invasion of just one nation. As a matter of fact, he did start his invasion with the Earth Kingdom, colonizing minor Earth Kingdom cities and provinces. With his forces, he then launched a massive genocide on the Air Temples as an attempt to destroy the Avatar Cycle.
The Southern Water Tribe was of no help due to periodic raids which devastated its population and removed the majority of its benders, while the Northern Water Tribe preferred to stay protected inside their walls, as they would be powerless without a limitless supply of water in their disposal. Therefore, in a way, the war itself is mainly against the Earth Kingdom. This is further proven in "Sozin's Comet, Part 1: The Phoenix King", when Katara says that the Fire Nation basically won the war when they took over Ba Sing Se, even though both Water Tribes remained out of Fire Nation control, an indication that the war was more with the Earth Kingdom than anything.