How difficult is Axis and Allies to learn and teach new players?
Moderate Difficulty

I assume you're talking about A&amp;A Revised or perhaps the 50th edition rules. I mention this because both are slightly more complicated in terms of rules than now obsolete 2nd edition (i.e. the new editions have more units and more special rules).

That said, the following observations should apply across all versions, with the newer ones being marginally more difficult. I'll address how hard it is to teach into a few key categories:

Rules complexity

Axis and Allies is rules intense. I would put the rules complexity about equivalent to Diplomacy. Consider it a full tier above something like Monopoly, Risk, or Clue, but not quite as bad as Arkham Horror. Furthermore, unlike a game like Dominion, where the base mechanic is fairly straightforward and most of the rules are revealed on the cards themselves (i.e. instructions at the point of need), in A&amp;A someone is going to need to have good grasp of the game from the start, or you're all going to be buried in the rule book for the first few games. Be prepared for the first game to take extra time.  

Strategy complexity

Although there is a steep learning curve between the novice and expert player, this should not be a problem for a play group learning together. There are optimal buys and well tested strategies, just like in chess. But if none of you know then you'll all learn together; years from now you'll all have a great laugh about the crazy stuff you tried when starting out. So yes, you need to play several times to get it, but unlike some games, I think A&amp;A is fun when you're learning, so it's not like you'll have to put in four or five games for it to pay off. 

Game geekery

I don't think you have to be a true game-geek to enjoy it. I've found its appeal is well calibrated with its rules complexity. Casual gamers who only like the most mass market of games will probably be intimated. But you don't have to be a hard core gamer who thinks Arkham or Twilight Imperium are too simple to like it. That is your group will need to want a challenge beyond the most common games, but being a geek not required. An interest in history in general, WWII, or war games helps as well, as the theme in this game is obviously very strong. If your group wish Risk wasn't so simple and boring, this might be a good option.

Family appropriateness 

Assuming your family meets the criteria above (ready to graduate to slightly more complicated games) then it might be appropriate. It's a long game, so your family will need to be willing to spend a few hours doing this (or have a big table to leave it set up between sessions). That said, the game is robust to people with different learning curves because it's a team game (Axis vs. Allies). You can team up the strongest and weakest players and/or give the best player the most challenging country. The game is also easy to handicap with bidding. 

I say go for it.