What's the best way to explain 7 Wonders to a new player?
I find that using the rulebook is often not the best way to teach 7 Wonders. There's a lot of text there. While a "here are your cards, let's go approach," can work well, most new players appreciate some form of idea.  I'll post the bullet points to my 7 Wonders Crash Course (given fairly regularly at game nights near me). Apologies for the impending wall of text.


This is 7 Wonders, in which you will receive a Wonder from an ancient Mediterranean civilization, which you must turn into a shining beacon of greatness.
To do this, you'll need Victory Points, which look like this (Point to a VP symbol).  Building your Wonder will get you some Victory Points, and a cool ability (explain their wonder ability).
You can get Victory points by playing cards.  Cards have a cost (Point to the cost of a card) and an effect (point to an effect).  Not all cards have a cost, but most of them do.
There are seven different types of cards, which do seven different types of things.  
First off, we have brown cards, which provide you with a basic resource: Wood, Stone, Clay, and Ore. (Have an example card on-hand for each)
Next, we have gray cards, which provide you with manufactured resources: Glass, Textile, and Papyrus. (again, have example cards)
Brown cards and Gray cards provide you with resources, which let you build other buildings.
Next up, we have Blue cards.  Blue cards give you Victory Points, the currency of winners! As you progress through the game, you'll gain access to buildings that provide more Victory Points. (Show some blue cards, preferably from all three ages)
Yellow cards are Commerce Buildings.  They mostly deal with gaining money, which you can use to buy some cards, and to get a discount on buying resources from the players sitting next to you.  (Show them Trading Posts, explain East vs West &amp; purchasing, and also show them a Tavern).
Red cards are Military Buildings.  Military buildings increase your Civilizations strength.  At the end of each Age (indicate Card backs), you'll compare your military score against your neighbors' military scores.  If yours is higher, you get Victory Points! If yours is lower, you lose 1 Victory Point. (Show the tokens, along with a red card).
Finally, we have Green cards.  Green cards are Scientific Buildings.  They allow your Civilization to unlock new and exciting technologies, which are really just Victory Points.  Science is complicated, but very rewarding, just like real life.  There are three symbols on Green cards, Cogs, Compasses, and Tablets. (Again, get some examples)  The more of any symbol you have, the better.  At the end of the game, you receive Victory Points equal to the square of the number of cards you have for each symbol.  So if you have one tablet, and two compasses, you'd get one point for the tablet, and four points for the compasses.  If you got five cogs, you'd get 25 points!  But that's not it!  For every set of three different symbols, you get an extra seven points!
We also have purple cards.  They don't come into the game until Age Three, so we can wait until then to worry about them.  The important thing is that they get you points for what people next to you have.


That was a lot, huh.  Don't worry.  It all clicks pretty quickly.  One last thing, and we'll get started.  You'll start with a hand of cards.  Each turn, everyone will pick a card to build, and place it face-down in front of them.  Once everyone has selected a card, we all turn them face up, pay any necessary costs, and put them by our wonder. If you don't want to build one, you can also discard it for three gold, or use it to build part of your Wonder.  Then, we pass our cards to the person next to us.  The hands keep rotating around until there's only two cards left, so you never know exactly what cards you'll be picking from on your next hand.  There's a handy chart for what all the symbols on the cards mean in the rules, so you can check it any time you want to, or ask the person that just gave you your cards, if you'd rather not.