Congrats on the iPhone. I believe it will continue to amaze and delight you as it has myself. I purchased one when they first came out, and I'm still blown away by what it can do, and what new capabilities continue to be added.
As far as applications go, there are SO MANY to choose from, and I have more than 100 installed. Most were installed on a whim, and haven't been used much, but a few essentials (for me anyway) have proven to be:
Dictionary - obvious
Facebook - pretty obvious
GPS - not free but great for tracking runs, hikes etc.
iConvert - convert measurements, metrics etc.
Kindle - read the Kindle e-books on your phone, way easier and cooler than you might imagine.
MailMe Text - instantly emails a note you might write to your email address.
Shazam - for identifying music being played (like in a restaurant)
ShopSavvy - barcode identifier. Finds that item and it's local pricing
Translator - English to Whatever, and Whatever to English
Tweetie - Twitter client
Then there are whichever News outfits you prefer, like CNN, Fox and WSJ, all have their mobile versions for you.
Plus, of course, I've added a direct link, that shows up like an app, to SBG.
But a big part of the fun comes from going to the App Store and just browsing, checking stuff out, and, since so many are free, downloading apps to try out.