1. Find My iPhone - This is hands-down the first app that should go on every iPhone, and it's frankly surprising Apple doesn't include it out of the box. This free app won't help you find your own iPhone, but if you live in a house with multiple iPhones and iPads, it will allow you to find someone else's.
If your wife drops her iPhone behind the couch and can't find it after several minutes of frantic searching, you can use Find My iPhone on your own device to ping her device and let her find it in seconds. And if your iPhone gets left behind at a bar and someone runs off with it, you can use Find My iPhone on your wife's device to track down the thief -- it happens all the time, and many people have recovered their expensive devices using this simple, free app.
2. Camera+ - For US$1.99, this app does everything the built-in Camera app ought to do. Camera+ takes photography on the iPhone to levels that seem impossible on a cellphone camera. With built-in exposure controls and grid lines, you have much more control over composing your photos. Once you've taken your photos, Camera+ gives you access to a tremendously powerful suite of editing tools that can make simple snapshots look like works of art by the time you're finished.
Camera+ has built-in sharing tools, but if you want a more social experience for photo sharing, there's nothing better out there right now than Instagram, which lets you share photos simultaneously across not only its own service, but almost every major social network out there. Instagram also has some basic photo filters you can use to make so-so pics stand out a bit more, but the real draw is seeing and commenting on other people's pics. It's very much like what Facebook does with pictures, but simultaneously more powerful and more fun.
3. Dropbox - Dropbox gives your iPhone the file storage solution that Apple still refuses to. While you can't use your iPhone like a hard drive or flash drive to store files locally (at least not without jumping through a lot of hoops), Dropbox is the next best thing. It offers online-based storage of files, with 2 GB of storage space for free. Even better, you can install Dropbox on a Mac or PC, too, which means sharing documents and files between your Mac and your iPhone is as simple as dragging the file into your Dropbox folder.
4. A messaging app - Why give your wireless carrier more money than you already do? Skype does video chat similarly to Apple's own FaceTime, but unlike FaceTime, it works over 3G (most of the time). You can also do free Skype-to-Skype audio with other Skype users, and there's a free (if basic and sometimes flakey) instant messaging feature as well. If you're already a Skype user and use the service regularly, Skype for the iPhone will serve your needs.
If you're not using Skype, or if you'd rather not use the service for one reason or another, BeeJiveIM ($9.99) is a much better alternative to Skype's instant messaging feature. While it won't do video chatting or voice calls like Skype, BeeJiveIM will allow you to instant message with your friends over a wide variety of services, including AIM, MSN Messenger, Yahoo! chat, GoogleTalk, Facebook, Jabber, and MySpace IM.
If you download both Skype and BeeJiveIM, you've got several communications alternatives to what your carrier offers. Instead of paying for minutes of talktime or SMS/MMS services, you can use Skype or BeeJiveIM to keep in touch for much cheaper.
5. Any Twitter app - Speaking of keeping in touch, Twitter is a fully viable replacement for SMS/MMS, especially if your client app has built-in push notifications that will allow you to get instantaneous updates wherever you may be. Again, why give your wireless carrier more money for overly expensive texting packages? Chances are if this is your first iPhone, you haven't signed up to Twitter yet. Signing up is free, and if you sign up and get your friends to sign up too, with the right Twitter app you'll be able to ditch your texting plan entirely.
There are so many Twitter apps on the App Store that it's probably difficult to choose one, but we've got some recommendations there, too. Twitter's official app is a good choice if you're just starting out with the service; it's free, well-designed, has all the basic features you'd expect, and doesn't bog you down with features so quickly that you get confused. Tweetbot ($1.99) is a good step up from Twitter's official app, especially since it has a very eye-candy and inspired design. If you're already comfortable with Twitter's basic features, Echofon is a nice, geeky choice for a Twitter app; it's not the prettiest of the Twitter clients, but it's one of the most feature-rich out there.
6. 1Password - Typing in passwords on the iPhone's keyboard is usually more of a pain than it's worth, and it's a good way to feel safe while you're doing mobile banking. 1Password allows you to store passwords for websites, bank accounts, Wi-Fi networks, whatever you may need. It's also got a great "wallet" feature that lets you input things you'd normally carry around in your wallet, like discount cards, library cards, or even your driver's license -- just don't expect the cops to accept your iPhone as legal ID if they pull you over. You can even jot down notes that you want to keep secured from prying eyes. If you want something on your iPhone, but you want to be the only one to have access to it, 1Password is what you need.
7. Instapaper - If you like to read articles online, and if you ever foresee reading them on an iPhone, you want Instapaper ($4.99). With just a bit of initial setup, you can configure a link in Mobile Safari to send articles to your free Instapaper account so you can read them later. Instapaper is an invaluable tool for creating a cache of reading material on your device. See a lengthy article that you don't have time for now? Send it to Instapaper and read it when you've got the time. Does part of your daily commute involve going through areas with no wireless coverage? Store articles in Instapaper before you leave, then read them later on the train. Instapaper's one of those iPhone apps that may not seem strictly necessary at first, but once you start using it, you'll never want to be without it again.
8. Evernote - With a simple interface that lets you make notes with text, photos, or audio, this is your brain on iPhone. This free app can also sync notes with its Mac-based counterpart, so you can type up a grocery list in Evernote on your Mac, sync it to Evernote's online servers, and have that information waiting on your iPhone when you hit the store. With Evernote on your iPhone, you never need to worry about forgetting anything again -- at least as long as you remember to look it up in Evernote, that is.
9. An ebook reader app - The thought of reading entire novels on the iPhone's tiny screen might seem ridiculous at first, at least until you try out either Amazon's Kindle app or Apple's own iBooks. If you've already got a Kindle, you can sync over any Kindle books you've purchased to your iPhone, and you can even sync your reading progress between the devices.
The only drawback to the Kindle app is that you have to back out into Safari if you want to buy more books. Apple's iBooks isn't saddled with that restriction, offering you access to a wide variety of titles in the iBookstore (though not nearly as wide a variety as what Amazon offers). Both apps are very competent ereaders, and while you won't get that book smell while you're reading, the ability to carry an entire library of books around in your pants pocket is pretty phenomenal.
10. Any game - The iPhone's a great communications and productivity device, but it's also got one of the widest games selections ever. Whether you're just after some casual fun or an in-depth, full-featured experience, you'll be able to find it on the App Store. You've probably already heard of Angry Birds ($0.99) by now, and that's as good a place as any to start your iPhone gaming experience if you're just after some pick-up-and-play fun. Plants vs. Zombies ($2.99) and Tiny Wings ($0.99) are other casual titles that are good for first-timers to check out.
If you want to see what the iPhone's really capable of, download Infinity Blade ($2.99 as of this writing, normally priced $4.99). The gameplay is a bit on the simplistic side, but it has the best graphics of any game on the App Store, with visuals that rival the quality of games on the PlayStation 3. If you find yourself exclaiming "This is on a phone?!" after a couple minutes with this game, you'll see why we recommended it.
11. TUAW - Okay, we fudged the headline a little bit: this list goes to 11. Our own app will get you the best Apple-related news anywhere on your iPhone. Everything you see at our site, you'll find in our app.
Keep in mind that this list is only the tiniest sample of what's out there for the iPhone. We left out a lot of great apps from this list -- particularly if you're in the US, there are several amazing apps out there that we didn't cover here. But the apps covered in this list will serve as a good jumping-off point for your experience with the iPhone. Above all, don't just stop once you've downloaded the apps on this list; part of the fun of having an iPhone is exploring what's available for it on the App Store.
We'll have more posts like this in the future with more app recommendations. Meanwhile, enjoy your new iPhone!