Living earth is phenomenal. I don't think it's free...maybe a buck or two. Until I found it I never used anything other than the stock weather app...or for more specific Wx forecast, the browser (I prefer iCab on the iPad 4--Mercury in the Mini) through weather.com. If you can swing the dollar bill though, Living earth is awesome!
As I mentioned, for a browser on the iPad 4--iCab is and has been ,u choice for three years. Awesome gesture support, an on board downloading set up, incredible third party support...Atomic is good as well, check out the free version on Mercury
I couldn't live without Early Edition. My morning newspaper. It's an RSS program the literally 'draws up' a morning paper each day. I use it every morning. During the day I use Flipboard for up to date news
For my magazines Newsstand from Apple and I subscribe to a half dozen rags through Zinio.
I like XPlane's simulators. There are a half dozen of them. I like the retro games on the Midway arcade (I'm 41

)...some of the sports games are cool...including Madden, Tiger Woods, and NBA Jam. If you're a soccer fan, FIFA is phenomenal. Tons and tons of cool games regardless of your interest. Diablo type dungeon crawlers, MMO type, RPG, racing, casual, casino...you name it, every board game we've played--I like monopoly and risk.
EverNote is invaluable. As well, it's been mentioned...the iWork suite; Pages, Numbers, and Keynote ;works with MS formats too: export or import .doc(x), xls(x?), and ppt. I use Bento daily for my business.
If you're the creative type and enjoy makin music check out GarageBand, the Korg synths, Nano Studio...Moog, the list goes on
Avid and iMovie are must haves if you shoot video with your iPhone. iPhoto and Photoshop of you're a photographer....this is just the crust. The list is really long with photography and video production applications.
Kindle, Nook, and iBooks. Pending where your collection resides. If you have a preference or digital book collection. If not, I prefer iBooks and Apple's prices are decent. Kindle has a huge library though (Amazon).
Congrats! You're going to love it! It's an amazing machine and the best one yet!
Jer