I think you should start with academy. This is a simpler game/faster than Arena and will be out very soon. Once you grasp the game and enjoy how it's played you could consider buying arena. 
Not necessarily. If you just want to learn how to play, you can just as easily do that with apprentice mode, no academy required! Whether you get into Academy or Arena or both really depends on what you prefer.
Arena is basically a minis game that uses cards instead of minis, it's been around longer so it has a lot more cards and mages released. And if you play in the demo mode a few times first it's really not too difficult. That being said, Arena generally takes 1-2 hours to play, and is more complex than academy. Zone movement is a pretty important part of the game.
Academy is more like a card game, takes 20-30 min to play, is simpler and has no movement.
Academy plays like a whole different game, but those significant differences in gameplay are caused by much smaller differences in the rules.
Academy is basically Arena, except with only 1 zone instead of 12, a different and less powerful card pool, different mages, and no planning phase. That might not sound like much of a difference, but it really really is. The thing about mage wars is that the rules have the potential to be kind of modular, and there are a lot of ways you could play with it.
I suspect the biggest factors involved in which one is best to pick might be time and money. Academy core set costs $30 on the Arcane Wonders Store and Arena core set costs $60, although they probably would be somewhat cheaper if you can buy them online somewhere else with free shipping.
Arena has a steeper learning curve, and it's better if you play in the demo mode at least a few times before moving on to the full game. Academy you can just pick up and play.
Arena has more strategic depth, but Academy is about as strategically deep as a typical collectible card game if not more so, so it might still be good enough for you.
Like I said, it's really a matter of preference.
Hope that helps!
