Making an enemy out of a neutral country when you are at war with many others means there is one more enemy to fight. So what was the plan of Japan or the Axis to drag US into the war? Did they think they could make the US a Japanese colony? If so how did they plan to win such invasion?
Neither Japan nor Germany intended to actually invade the United States. Instead, they saw the US as a threat to their plans for conquest of SE Asia and Europe.
Japan and the US saw each other as rivals for control of the Pacific. They had the largest fleets, and overlapping interests. The US had its island territories won in the Spanish-American War, primarily the Philippines, it would protect. Japan, increasingly isolated by the international community for its violations of treaties and war in China, was looking to secure natural resources. Tensions were high.
With its vital imports cut off by embargo, Japan planned an invasion of the Dutch East Indies and other European colonies. They knew such a move would touch off war with the United States and decided on a pre-emptive strike to destroy the US Pacific fleet at Pearl Harbor while invading all US and Allied possessions to deprive them of bases. The hope was not to defeat the US completely, but to demoralize her and destroy her ability to project power. By the time the US fleet had been rebuilt, Japan would have a fortified ring of defenses and could negotiate a peace. That was the idea anyway.
Germany's motives remain unclear. Germany had long been annoyed by the US's increasingly open support for Britain including arms, food, and ships. US destroyers sparred with German submarines even while neutral. But even while the US government was mobilizing and becoming ever more allied with Britain, US sentiment was against war with Germany.
Shortly after Pearl Harbor, Germany declared war on the US. This was done by Adolf Hitler without consulting his advisors. Numerous theories abound as to why, the strongest is he hoped Japan would in turn declare war against the Soviet Union. Japan, having abandoned its northern strategy of expanding into Siberia, still fighting a war in China and now against the United States, and still smarting from a humiliating defeat against the Soviets at Khalkhin Gol declined.
Declaring war against the US was probably Hitler's biggest blunder. Faced with the threat of Japan, the US likely would have thrown it's might into that fight and ignored the European problems. Instead, Germany was now at war with an enemy it had little ability to strike at, though there was about nine months of slaughter as German U-Boats ran rampant over the ill-prepared US coastline. US industry, manpower, and warships could now freely support the Allies.
Germany and Japan had some ideas about how to strike at the US. Few were practical, even fewer were actually built, even fewer would have affected the war.
The Japanese had the I-400 class submarine. Each carried three small aircraft. The Japanese planned to build 18 and use them to strike the Panama Canal and prevent the US from moving ships to the Pacific. This might have had a strategic effect on the war had they been used early enough. Instead, they built 3 so late in the war they were never used for their intended purpose.
Germany had the Amerika Bomber, a long range bomber intended to attack New York City. It was overambitious and as the war progressed Germany lacked the resources to develop such an aircraft. The program was scrapped.
Ian Kershaw's biography of Hitler has some interesting insights (pp656-660 of the single-volume edition).
Hitler felt that the Japanese would tie down the US in the Pacific, and weaken the British by attacking their Far East possessions. He was also delighted to have "an ally which has never been conquered in 3000 years." He had given the Japanese a verbal commitment on 4th December 1941 that if war broke out between Japan and the USA, Germany would consider itself at war with the US too, and Italy had willingly joined this new tripartite pact. That was completed on 11th December as a commitment by all parties not to make a separate peace with the USA.
Hitler felt that FDR was looking for an excuse to intervene in Europe, and the general German opinion was that if the US was at war, there would be no more Lend-lease for the British: the US would want all its war production for its own use.
He formally declared war on the US in a speech to the Reichstag later on 11th December. He felt that this formalised what was already the effective situation, and demonstrated that he was still in control of events. Waiting for a declaration of war from the USA would have been a sign of weakness, in this thinking. His primary target for this speech was the morale of the German population; this was a way to get them to view a new opponent as a positive development.
He underestimated all his opponents, the US most of all.
The Axis countries attacked the United States as a "deterrent." Take Japan at Pearl Harbor. The intent was to "knock out" the U.S. fleet for two years, during which Japan planned to conquer Southeast Asia (which she did) plus China and India (she came fairly close). The expectation was that by January 1, 1944, Japan would be confronting the U.S. with a behemoth consisting of Japan, China, India, and today's "ASEAN" nations, and the U.S. Would have no choice but to make peace.
What Japan didn't count on was American resilience. Six months after Pearl Harbor, a "weaker" American fleet knocked out four Japanese carriers (about half of Japan's striking power at Midway, while suffering only one quarter of the losses (a single carrier). Combine this with Japanese losses at Coral Sea and Guadalcanal, and the Japanese fleet was basically "done for" (it had one last hurrah in waters around the Philippines in mid to late 1944). By mid-1944, the Americans had "island hopped" the central Pacific from "Tarawa to Tinian," the latter on Japan's doorstep, effectively separating the Japanese islands from her Asian possessions.
Then Germany's Admiral Doenitz felt that he could knock out America's merchant marine by sinking ships faster than they could be built. He was correct for 1942 that he could sink faster than we built, but the Americans still managed to land a large enough force in North Africa in November, 1942 to draw away enough German supplies and reinforcements from the battle of Stalingrad so that the Russians could win, followed by an invasion of Italy in 1943. (The Axis had not expected America to make its weight felt until 1944, by which time Germany expected to have conquered Russia, and possibly Britain as well.)
The Axis tried to "buy time" with pre-emptive strikes on America, but given the speed and severity of the American reactions, and the holding out of America's other allies, they didn't buy enough time.