If you ever have to restore your iPhone, I highly recommend setting it up as a new phone -- especially if you sync your mail, contacts and calendars over the air with Mobile Me.
While it isn't nearly as convenient as restoring from a backup, setting up as new removes a lot of the crud that can build up on your iPhone, especially of you install a lot of apps (or if you're a jailbreak aficionado.)
The main problem with restore/set up as new is app data -- if it's not synced to the cloud, it's often gone forever. Before going this route, make sure that you back up your photos. Photos aren't backed up automatically (unlike most other aspects of the iPhone) and a restore will delete them.
After you've restored and want to set up your iPhone as new I recommend just installing the core apps that you use on a daily basis and adding the rest incrementally -- as needed.
Following is a list of my favorite iPhone apps:
Front Page apps
Google Mobile App (free, App Store) – Quickly search Google using your voice and location. A feature sorely missing in iOS.
TweetDeck for iPhone (free, App Store) – My favorite Twitter client. My review (November 2009).
Twitter for iPhone (free, App Store) – My second favorite Twitter client.
Foursquare (free, App Store) – While I'm not a prolific check-in app user, I do like using it while traveling.
Skype (free, App Store) – Unbeatable way to make free/cheap calls from the iPhone when traveling internationally (although I prefer video iChat when on OS X).
Light-O-Matic ($0.99, App Store) – My favorite flashlight app for the iPhone 4.
Accuweather Web app (free) – Yes, my favorite weather app is a Web app. Bookmark http://apple.accuweather.com and save it to your home screen.
The Weather Channel Max ($3.99, App Store) - While I prefer the compact interface of the Accuweather Web app, I like the GPS feature and 10-day view in TWC Max.
AyeTides ($9.99, App Store) – Simply the best tide app out there. If you need to know that tides in your area, this app is for you. My review from 6/27/2010.
Surf Watch ($9.99, App Store) - Beautiful app that monitors the wave heights at the NOAA buoys and has fully configurable push notifications and alarms that warn you when big waves are coming. Sweet!
Surf Report (free, App Store) – This one gives you the current wave heights and ocean temps for your favorite surf sports. Use its GPS to find new surf spots when traveling.
Pulse New Mini ($1.99, App Store) – Killer, visual newsreader for the iPhone.
Reeder ($2.99, App Store) - Google reader client app.
Productivity (a generic catchall I use for several types of apps)
1Password Pro ($9.99, App Store) – The best password manager for OS X comes to iPhone and now has automatic syncing via DropBox. (My review from November 2009)
TextExpander ($4.99, App Store) – Type faster on your iPhone with text snippits that expand to larger words/phrases.
Dragon Dictation (free, App Store) – Fills a hole in Apple's missing voice recognition, uses the widely praised Dragon NaturallySpeaking engine.
AirMouse Pro ($1.99, App Store) – My must-have application for controling VLC on the Mac mini media center in my living room.
PasteBot ($3.99, App Store) – Clipboard manager for text and photos.
DropBox (free, App Store) – 2GB of free could storage. Indispensable on OS X and on the iPhone.
Pandora Music (free, App Store) – What else is there to say? Should be part of the default iOS 4 installation IMHO.
Live Phish ($1.99, App Store) – Innovative streaming audio app from the band Phish. Access to all of your purchased music and streams a random free show -- now with background audio. My review from June 2010.
Midomi SoundHound Infinity ($4.99, App Store) – What's that song? This app will tell you.
TWiT ($0.99, App Store) – While technically not music per se, I store my favorite podcast app in my music folder for lack of a better location.
That's my short list. What are the first iPhone apps that you install?