Not as a way to brag, but rather to establish credibility, I have been first and second respectively at the WI and NC Opens this year in extemp... so at this point, although I don't really feel like it, apparently I kind of know what I'm doing
The most important thing you can do is have a deep and broad knowledge of current events, knowing what questions are talking about without having to do research on the spot is very helpful. If you already follow the news you're in good shape, if you don't, well, that's okay too, extemp is awesome.
If you're willing to pay for it, get Extemp Genie, it's $25 dollars for a year and worth much more than that. It saves so much time collecting articles and you will never find yourself stuck with a topic you have no articles on. For the future a subscription to The Economist
is also a worthy investment, they have excellent coverage of worldwide news delivered to your very own mailbox every week.
Okay so enough with spending money on things. The first step after draw is to select the question you want to talk about. Judges will appreciate if you choose a topic that is different and interesting, but the most important criteria should be which question do you know the most about already. Unless you have superhuman qualities, you will always speak better on a topic about which you are knowledgeable and/or passionate.
Once that's done decide on your answer to the question. Practically everyone follows the same basic pattern for an extemp speech: a hook, read the question, read your answer to the question, roadmap your three points, point one, point two, point three, re-read question and your answer and sum up the three points and how they support your answer, tie your hook back in if possible. It is great if your point's taglines can be rather catchy. A lot of people use some version of "past, present, future". It's simple and easy to remember (and lets you sneak in some interesting background information that may or may not be really relevant to the question). You can also use alliteration, a common sentence structure, or rhyming to make your taglines stand out. With the basic structure and content of the speech in your head, you can start really writing it, using Extemp Genie to find articles that support what you're saying. The articles do not always need to directly say what you are saying, for example, suppose you want to say that Hillary Clinton is seen as the inevitable candidate, but you can't find an article that says "Hillary is seen as the inevitable candidate". In that situation, I instead found an article saying that Bill DeBlasio's endorsement of Hillary was inevitable, mentioned that fact in my speech, and then said that that was exemplary of the key word "inevitable" in Hillary's campaign. So essentially my strategy is to mentally build the generalities of the speech and then find evidence to support it. Others find it better to find evidence first and then use it to shape the speech. You'll have to work out what's best for you.
Having humor is very helpful. Most speeches of mine, and of others, are really very boring. Having a really good opener and scattering some jokes throughout will make you stand out. Another tip is to fold the 3x5 card in half. That way it's smaller and less obtrusive. But most of all have fun, extemp is my absolute favorite category for a reason. It's like skydiving for nerds. Your debate experience will help you out a lot. Don't let the prep-room proctors scare you too much. Remember to turn off your laptop's WiFi in prep. Label your computer. Your first speeches will probably suck, but it gets better. Practice is invaluable. I hope this extensive ramble is at least somewhat helpful.