In addition to making the world Brave and New, Civ 5 's latest expansion can also make it a little confusing at first. With the huge changes to trade, culture, and diplomacy, you'll probably want to know a few things before you start shepherding your next civilization to victory. Listen closely, my liege...
Before Brave New World, the main way to boost your income for most of the game was with buildings and, sometimes, tile improvements. Now, the new trade route system is what will be filling the lion's share of your coffers. If you want to get rich quick, research caravans (and later, cargo ships), and max out your trade route limit as early as possible.
Also keep in mind that trade routes are more profitable the more resources each partner has that the other doesn't. A wise Trade Prince will want to found cities on scarce luxuries for maximum profit. And don't forget to protect your trade routes from pirates and barbarians with a decent army and navy.
The new method of culture victory involves overpowering the Culture of all other civs with your Tourism. If you're going for a Culture victory, this makes it important to get your culture train rolling as early as possible. But you were already doing that to stack those amazing policy bonuses, weren't you? Weren't you?!
If Culture victory sounds like your thing, be aware that you won't be able to generate a lot of tourism early on. Do whatever you can to boost your rate of Great Person generation, and rush like mad for the wonders that can hold Great Works with excellent theming bonuses. You will need to do some trading to make the most of said bonuses, but be sure you don't trade away something you're going to need for another slot later.
The World Congress becomes possibly the deciding factor for the rest of the game from the moment it's founded. Other than enabling Diplomatic Victory, it allows you to cripple your rivals by non hostile means... though they can do the same to you. It can give you a leg up on other victory types with shared wonders and mandated research funding. Ignore it at your peril.
By extension, City-State allies are now even more important, as they come to represent a majority of the possible votes in the World Congress as the eras draw on. I haven't played a single game of Brave New World in which I didn't end up taking the Patronage policy tree to exploit this. Be nice to the little guys, and their votes will lead you to dominance.
Ideologies (Freedom, Order, and Autocracy) have changed a fair bit. You now have an incentive not to join the prevailing world ideology, as the first civ to choose each branch gets two, free policies. They also now tie into the tourism system. Civs with high Tourism can cause unrest in civs using a different ideology where their culture is popular.
Also beware power blocs with lots of city-state allies, as they can pass a resolution in the World Congress instating a “World Ideology,” making it very hard for anyone outside that ideology to get anything passed ever again. A similar resolution exists for World Religion.
With the above suggestions in mind, you should be prepared to face the Brave New World that awaits you. Oh yeah, and Alexander is still a huge jerk.