How can I speed up a game of Axis and Allies?
This is an excellent question!

For a board game I love but never play now since me and my friends, all working, can never get the 50 hours needed to finish it :) (ok, it's more like 8 hours no kiddin') if we don't simplify the rules and/or shorten the terms.

We have successfully used gentlemen's agreeements, as follows:

Shortened victory conditions (classic, revised, 1942, anniversary edition and 1940s):


We use the shortest victory conditions, always. It still takes many hours
Capital conditions: If a capital falls; the game is over even if it can be retaken. In practice that is a game winner anyways. I am sure there can be exceptions, but not if you play aware of that condition. It affects mostly the crucial Russia/Germany stalemate.


This helps a lot since you may reach a point at which Germany has good chance of taking Moscow in two rounds, or die so you can simply resolve only the moves and battles that affect that and call it a game. 

Japanese stalemate settlement (classic, revised, 1942, anniversary edition and 1940s):


Our housee rule is: If japan has been wiped out from the mainland and has lost its fleet while there are at least scattered Allied ships in the Pacific, give all VP cities to the Allies and check if game is over.  Taking Tokyo is a 6 round affair at this point, that takes hours.


Europe and Pacific 1940:


For Germany: if USSR is advancing while the Atlantic has protected Allied transports unloading troops in Europe for two rounds or more, we call it a game. (Even if Italy is huge, because it failed to counter the previous.)
For Japan: If Japan land troops are both in Caucasus and eastern Russia (3-4 territories) while Japan holds the biggest Pacific fleet, we call it a game as well. 


If you are new or almost new to the game:


I think turn time limit helps a lot. Agree on one, just make sure Germany and Japan have a bit higher limit than the allies
If you will play often, play at least 3 games of limited rounds (for instance, 3). Discuss who would win and why. For games like 1942 or Revised this helps a lot because you will better understand what can be achieved at that round and will have an idea about how long your games will take.


And the golden house-rule: Accept the dice, be a gentleman.

Agree on these kinds of conditions when the game starts and accept that the game is over at any of these points.  

I am serious on this, because most games finish with a bitter taste of no conclusion with statements like "but I could still attack here and win that and..." which mostly leads to hours of playing to reach the same apparant result.

Not surprisingly there are these A&amp;A chess and "Low luck" (tripleA videogame) housee rules that to me are based on this bitterness. The game is not designed for that, thus making the game somewhat scripted after a while.