Axis and Allies seems to be a very popular game to the point where I've even seen copies in Toys 'R Us! Before I jump on the bandwagon, I want to be sure I'll be able to get some use out of it.
I assume you're talking about A&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&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&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.
How difficult is it to teach new people? Is it the kind of game you need to play several times before you really get it?
Axis & Allies is a turn-based game. When it's a player's turn, she goes through a series of steps to complete her turn, after which it is the next person's turn. There are a number of distinct tasks that can be performed during a turn and mastering that sequence takes at least one detailed reading of the rulebook along with a fair amount of practice time. But once you have the turn mechanics mastered, the game moves very quickly.
So to answer your question, if you are well-versed at the game it can take a little while to get a new player up to speed. It also requires a bit of patience because it will take the new player some time to get in the groove and it will take a bit of time for them to complete their turns!
Would it be a good game to play with family?
If you have family members that enjoy lengthy, turn-based games, then yes. My family enjoys games, but more along the lines of card games or fast-paced board games. Spending 15-30 minutes setting everything up, just to start playing is not something any of my family members would subject themselves to!
Do you have to be a Game-Geek to have a chance at enjoying it?
I don't think so. I love the game and I'm not remotely close to what you'd call a board game geek.
Before you invest in the board game, it might be worthwhile to first read the rulebook and then try the game out online. That way you can get a feel for the turn-based mechanism and see if it's a type of game you (or your family) would enjoy.
You can play the 2nd Edition of Axis & Allies online (for free) at http://gamesbyemail.com/Games/WW2. (It's not an interactive game - rather, you make a move and then your opponent is notified. He can then make his move at his leisure, after which you'll be notified.) And you can read the 2nd Edition rulebook at http://www.wizards.com/avalonhill/rules/axis.pdf. If you're interested, drop me a line and I'd be happy to play a game with you and offer any advice or answer any questions you have during our play.
Granted, playing online is no substitute for playing it in-person, as you lose the social dynamic, but the nice thing about online play is that you don't have to commit 15-30 minutes for setup nor do you have to set aside a big block of time to play the game start to finish.
It's not that difficult to learn at a base level, but the large ruleset, range of move variations and many pieces definitely needs everyone's attention and interest from the start.
The first time I tried with a couple of 'game geek' friends, we spent an hour and a half setting it up and reading the instructions, only to realise we'd never finish a full game that evening anyway. Set aside time for a good few trail runs.
I see no reason why it wouldn't make a good family game, maybe not so much with some younger kids due to it's complexity.
One persons geekery is another persons passion but it is definitely is at the more complex end of the gaming spectrum. It would also possibly help if you are a fan of war games already - can you tell your British Cruisers from your Japanese Battleships?!
If you haven't already, it might be worth trying Risk, or Diplomacy first (the latter is my personal all-time favourite).
This is not an easy game to pick up. It takes several hours to learn, or at least learn well. It's multifaceted, with diplomatic, military, and economic considerations. Then playing could take the better part of a day. Best left for a long weekend or say, cruise vacation.
It would be a good game for MY family. But that's because we're all game geeks.
