The thing about Axis plans for grand conquest, perhaps funny if you think about it, was that there was no grand plan; only a continuing series of escalations until the most of the world's armies were at war. Germany, Italy, and Japan didn't pre-plan or coordinate much of anything when it came to the grand strategy to achieve all their expansionist objectives, but were rather flying at the seat of their pants as each nation bit off much more than they could chew.

Lets take Germany first, which already had a history of expansion through Austria, the Sudetenland, and Czechoslovakia without a single shot fired. When German leaders planned the invasion of Poland, some honestly believed that Britain and France would not come to Poland's aid, just as they didn't for Austria or Czechoslovakia, but were prepared to accept that the conquest of Poland meant war with Britain and France. This was seen as an acceptable outcome, as Germany and the Soviet Union recently signed the Molotov-Ribbentrop Pact, protecting Germany from the possibility of fighting a 2-front war, and proceeded with the invasion of Poland once the Poles rejected the ultimatum to cede Danzig.

During the months between Poland's fall and the invasion of France, Germany spent months with their soldiers in the factories building tanks, artillery, ammunition, and engines in preparation for a long campaign against France rather than march out on campaign. All of the German Army's high command (That I'll call the OKW from hereon out) and Hitler himself believed that a campaign against France would take months, even up to a year and feature a repeat of the trench warfare that was prevalent during WWI. When the invasion of the Low Countries and France finally commenced in May of 1940, the Germans, French, the British, and the entire global community was shocked by the rapid speed of Germany's advance, with the French military command collapsing and the nation signing an armistice in just 6 weeks. Germany's victory was wholly unexpected by everyone and changed the war entirely, with the global public nearly convinced that Germany was going to win the war. Britain refused to surrender however, and kept up the fight in the skies over London and in North and East Africa. During the relative lull in the fighting after the Battle of Britain, Hitler and the OKW began planning Operation Barbarossa, a further escalation of the war in invading the Soviet Union, which was Hitler's ultimate foreign policy goal.

The timing of Barbarossa was impacted by the perceived weakness of the USSR's Red Army after a poor performance against Finland in the Winter War due to a purge of the Army's officer corps by Joseph Stalin. Hitler and the OKW knew that the Red Army was in the process of reorganizing and rebuilding after the Great Purge and would be too powerful to defeat if left alone, so Hitler decided to push forward with the planning of the invasion while Britain was still in the fight, because waiting any longer would mean that the Red Army would no longer be weak enough to take on. Hitler was also convinced that as the war continued, Soviet or American intervention in the war was inevitable, so why not knock out the Soviets first (also as an aside, do not take this logic as a preemptive attack on the Soviets that some claim by reading Viktor Suvorov's "Icebreaker". Preventive War is more accurate than preemptive in this case). Barbarossa commenced with Germany advancing at lightning speeds just as was done in France, but Germany was ultimately defeated at the Battle of Moscow. Hitler lost on his gamble to defeat the Soviets and committed Germany to a 2-front war against the Soviets and Britain, soon to be with the United States as well.

For Italy, Mussolini corresponded with Hitler once he received word that an invasion of Poland was imminent, asking if Germany was intending to call on Italy to enter the war as per the agreement made in the Pact of Steel. Hitler informed Mussolini that he had no intention of calling Italy into the war, which suited Mussolini just fine. His army's staff informed him and the Germans that due to Italy's military activities in Ethiopia and Spain throughout the 1930's, Italy was wholly unable to fight a large scale war due to depleted equipment reserves, and needed until late 1942 before the Italian Army was fully equipped and ready. Mussolini was more than happy to sit out the first few years of the war and join when the army was ready; but Mussolini, as well as the rest of the world, was blindsided by the rapid collapse of France. Seeing a German victory in the war was now inevitable and wanting to obtain a slice of the victory pie before his opportunity was gone, Mussolini declared war on June 10, 1940.

Mussolini scrambled to declare war so fast that Italy's merchant fleets weren't called back into harbor in time and were still out at sea, making them open game for the British Royal Navy immediately. Very little to no supply or logistics arrangements were made for the Italian armies told to advance into the French Alps and Egypt, but for Mussolini that didn't matter. What mattered to him was that at least some territory was taken and some Italian blood was spilled to earn Italy a seat at the table once negotiations for the end of the war started. Mussolini rather famously stated "I only need a few thousand dead so that I can sit at the peace conference as a man who has fought." Mussolini entered the war expecting the conflict to be in its closing stages, but the war began to escalate in scale. Hitler informed Mussolini of the invasion of the Soviet Union only 6 hours before it was launched, giving him no time to object offer counsel, but stating that Germany does not expect Italian armies to participate in the campaign, but they were eventually called in to participate.

Now lets get to Japan, which has already been embroiled in a war against China since 1937. As the war with China dragged on, Japan came under increasingly harsh economic sanctions by the Western powers, including shipments of coal, scrap metal, and oil in order to levy Japan to give up its campaign in China. After Japan annexed French Indochina after the Fall of France in 1940, the United States placed a total embargo of trade with Japan.

With these resources now largely cut off, Japan was staring at a major resource crisis that would leave its army unable to fight effectively. Rather than retreat from China and admit defeat, Japanese military factions conspired for alternative ways to gain the resources they were cut off from and continue the war. The Japanese Army high command favored a northern expansion doctrine, Hokushin-Ron, which called for expansion into the Soviet Union and seize the resource riches of Southern Siberia. This plan was played with through a series of border clashes with the Soviet Union and Mongolia in 1939, but the Soviet Union wholesale defeated the Japanese border army at the Battle of Khalkin-Gol, causing Hokushin-Ron to fall out of favor with the Japanese high command.

The Japanese Navy instead pitched a Southern expansion doctrine, Nanshin-Ron, which called for expansion into the South Pacific to gain the oil fields of the Dutch East Indies and mineral rich SouthEast Asia. This plan would mean certain war with the United States however, which held possession over the Philippines. The Japanese Navy then came up with the plan to attack Pearl Harbor. The plan was to destroy or disable so much of the American Navy at once as to enable Japan to conquer the entire south Pacific without any US ability to respond, put as much physical distance between the US and Japan as possible, and bring the US and by extension Britain and the Dutch to the negotiating table and enforce Japanese hegemony over East Asia. Pearl Harbor was certainly a powerful attack, but not nearly as devastating as they hoped, as the US Navy recovered within 6 months to defeat the Japanese at the Battle of Midway.

To conclude, the Axis weren't out for total world conquest, but were after de-facto domination of the world's political stage by launching simultaneous wars of aggression against the Great Powers of the world, in a series of escalations that would keep compounding until the Axis were clearly outmatched in all aspects by the enemies they made. The Axis however weren't truly an alliance with a common cause with integrated coordination, more like 3 powers that went to war with common enemies, and distrusted each other almost as much as they distrusted their own enemies, leading to complete failures in coordination and communication between the Axis. Eventually in Europe, all of Germany's axis allies would all eventually just lose their independence to be militarily and politically dominated by the Nazis, rather than establish trust and communication.

Sources:

The Second World War by Antony Beever

The Origins of the Second World War 1933-1941 by Ruth Henig