The Soviet Union
The Soviets start the game in a tough spot--tantamount to their actual position in 1942--and are generally considered the weakest of the three allies. Attacking Finland/Norway or Eastern Europe on the first round can end the game quickly, but their typical goal is to delay the German attack for as long as possible to give the U.S. and Britain a chance to come to their aid. Experts at AxisandAllies.org suggest spending money on infantrymen if you wish to adopt a defensive strategy. Purchase eight men and move all of them save one from Russia to Karelia, along with the Soviet planes. The one man prevents a German blitzkrieg while presenting a formidable defensive wall in Karelia. By the time the Germans burst through, the other Allies should come to Russia's aid.
Germany
The German strategy hinges on wiping out Russia as quickly as possible, then turning to the remaining allies. AxisandAllies.org recommends purchasing four tanks and one fighter in the early rounds, then devoting the German fleet and air force to destroying the Allied shipping in the Atlantic and the North Sea. The ground forces can then be devoted to attacking the Soviets, striking at Eastern Europe, Karelia or the Caucasus territories. If any troops or fighters can be spared, Egypt makes a tempting target as well.
Great Britain
The British want to focus on Germany as much as possible. Air power can accomplish a number of goals fairly quickly, though if you want to plan for a long-term game, you may want to build a fleet instead. With airplanes, focus on building bombers, while sending the existing British fighters to Karelia to reinforce the Soviets. Strategic bombing runs can reduce Germany's production capacity, while troops in Africa can be sent to India to reinforce against a possible attack from Japan.
Japan
The temptation for Japan is to launch an attack against the U.S. naval fleet. This is usually a mistake, since the U.S. can absorb those blows and by the time Japan shifts gears, Germany is usually done for. Instead, you should turn your forces west and concentrate on wiping out the Soviets. Sweep into Eurasia with as many forces as possible, creating a second front for Russia to deal with. Once the Soviets are destroyed, Germany and Japan can protect each other's back while focusing on the remainder of the allies.
The United States
As the "arsenal of democracy," the U.S. works best by helping Britain and the Soviets take care of Germany. While you want to create enough troops to block a Japanese invasion, the bulk of your resources should go into bombers and transports, with any remaining funds used for infantry. The U.S. can defend Africa with troops much more readily than Great Britain, allowing the British to hold India more readily and deny Germany any resources in Africa. Strategic bombing runs can drop Germany's production capacity, starving the Nazis out and letting Russia and Britain finish the job.