I just ran my first game of Apocalypse World. Before that, I played in someone else's AW game that ran about 10 sessions. Here's my advice.
Read the book. Cover to cover. Don't skip stuff, even if you think you can get by without it. Basically, you should be familiar with the rules so you are aware they exist and can reference them if needed in play. I ran out of prep time and stopped reading after the "First Session" stuff and regretted it.
Listen to Vincent. Throughout the book, he tells you exactly how to run the game. Pay attention. Most of the advice I could give you, he already gives you in the book. Trust the process.
Get the players on the same wavelength. Read the intro to them. Discuss the apocalypse a bit and the psychic maelstrom. Talk about sex moves. Tell them that their job is to play their characters as if they're real people. Explain what your role is as MC.
Print out your MC moves. Refer to them throughout play. You're not just running the game freestyle. You have a list of principles to follow and moves to make. Play the game as written and eventually it will become second nature and you can "freestyle" that way. Luckily for you, there's a reference sheet on the website (it's at the back of the character playbooks PDF). (Also realize that the playbooks and stuff are legal-size, so you don't discover at the last minute that you don't have the right paper.)
Engage the game emotionally. The game deals with tough topics like survival, violence, and sexuality. Don't laugh these off or play them lightly. Engage them with all of your senses. Don't go for senseless gore or f**cking, but portray the human side of these things. Remember, one of your jobs is to play the NPCs as if they were real people--not cartoon characters. In the game I ran, I probably laughed off too much of the violence and so it became a little over the top and slapstick. However, the sex scenes that the characters starred in were realistic, emotionally engaging, and hot. Sure--a little awkward at the table, but you stop before things get really awkward. Just go as far as the group is able. You'll be surprised what your players can handle if you don't treat it as a joke.
Ask questions. Vincent already includes this as a move, especially as a First Session thing, but really do it. Use questions when you're stalled as MC and when you're wondering about some part of the world and what the players think about it. About half the time a player asked me about something in the world, if the answer wasn't clear to me, I threw it back at them (or another player). Questions are a cool way to guide players gently through difficult scenes, too. If they're getting giggly about a sex scene, ask "Is Killjoy giggling? How does Killjoy feel about seeing BlueLight naked?"
Good luck with your game!