I do not know the order of importance, and I will not try to order them. I consider them roughly equals.
Agreeing with Lucifer I consider buying am important factor. What I can do is related to what I buy, and I will buy after the current turn. Moreover buying is related to the IPC I make and the IPC I think that I will earn. Buyng unit you can not afford is hampering for your srategy. Buying unit that do not help you in delivering your punch where is it needed is also bd.
I think that Logistic is important. Having a costant flow of units going to the front is fundamental. Battle are fougth every turn, losses have to be replaced. Units to replace losses had to be buyed (see preceding point) and sent to the froint timely (before the losses reqiures their presence)
Tactic and mechanic is important. I mean here the operative possibility of the units. Knowing them helps in better positioning and better using of them. For example: German Fighter in WE and EE may help in trading territories on the russian front (Karelia, Belorussia, Ukraina) and then come back switching their position (those from WE land in EE and those from EE land in WE) ready to defend those terriotories in the remain of the round. Blocking a fleet with a lone TRN may allow for saving a territories from an invsasion.
Probabilities evaluation. Math! I think that making calculations about attack and defensive strenght may help a lot in taking soundly decision. Probabilities may help in taking decision.
Luck. Being A&A a chaotic system slightly difference in the outcome of the battles in the initial rounds will determine completely different evolution of the game. Butterfly effect.
Imagination. Being creative may disorientate our “usual” opponents. If the way in wich games are conducted is similar then yo uare predictable and this may create problems.
I do not know how to define this in englis. In italian we say “colpo d’occhio”, i.e. the ability to see at first look the situation being pushed towards a decision. Is something bounded to intuition. Sometimes, happend to me that I see an opportunity on the board only after I finished my combat, and so I can not exploit it.
Concentration. A&A turn is fought with a particular sequence (buying, moving, combat, NCM, deploying). The final situation of our army on the board is the result of all this phases. Being able to “predict” the final configuration is hard and require evaluation of probabilistic outcome, movement, missed possibility and exploited ones. It helps a lot.
Experience. The more I play the more I am aware of when I am making a mistake, when I make a weak move, when I make a strong move. Moreover I am able to see errors of my opponent that I can not see in my firts game and I am also starting to be able to exploiting them.
A good A&A player should have good in this features. The more features he is able the more he is good at the game.
I have not mentioned Strategy. Why? Because it is the sum of this concrete skills and of our abstract idea of how we would like “to win” the game.