The first thing you should get is App Shopper. It will filter all of the App Store and let you know what is on sale, the price history of the app, new versions, etc. I basically don't buy anything unless I check it out on App Shopper, because the App Store is a wasteland where you can never tell if something is on sale (yeah, the App might say 66% off, but App Shopper will show you that it was cheaper before, and likely will be again).
Off the top of my head, the most used ones I have are probably Dropbox, App Shopper, Words with Friends HD, one of the plain text writing programs (Plaintext, Writer, etc.), Netflix, ESPN Scorecenter HD, Puzzle Quest 2, some free PDF reader I found with App Shopper (I hear GoodReader is the best).
Really though, I buy all sorts of stuff when it gets cheap on App Shopper or snipe it when it goes free for a day.
I'll breakdown some of my favorites that I use on a daily basis:
Utilities:
1Password: This is an absolute must have. And works across platforms, so on your Mac, and if for some crazy reason you have a windows machine, even on there.
CalcBot: The best calculator for iOS. It's also a universal, so it'll work on other iOS devices (iPhone / iPod Touch)
Stash Pro: Best way to keep your documents (images mainly) secure from prying eyes.
Graphics:
Photogene: A very good image editor. This isn't Photoshop, but it gets much of what I need on the go done.
GPS:
MotionX GPS: Very inexpensive, great GUI, and invaluable if you have the 3G model that includes GPS.
Games:
Fieldrunners: This game is super addictive with great game play and graphics. It's a tower defense game.
Reading:
iBooks: I love this app, constantly am reading with it.
Flipboard: Excellent way to read news, very very nice interface.
Instant Messaging:
Beejive: Works well, and has a nice interface.
News:
Weather+: very slick graphics.
Weather HD: very slick graphics, especially if you want to show off the iPad.
Engadget: Self explanatory.
Air video is great.
I use atomic browser for tabs but it's not as responsive as I would like.
ibooks is a must if you read. I use it to get my large library of books on the ipad. If I were going to buy from a store i'd probably use the kindle app.
MotionX Drive HD for GPS if you need it.
I use feeddler for my google reader feeds. My most used app.
I have the official twitter app.
I use filer and dropbox for downloading files on my ipad and sending them to dropbox
ABC Player (Watch ABC shows for free on your iPad)
iBooks, Kindle, Nook, or Google Books
Pandora (streaming music)
Netflix (for movies)
Atomic Web Browser (IMO faster browser than Safari with desktop style tab UI. Doesn't refresh your tabs! Gives you a low memory warning instead when you have too much open. So much better than Safari)
WoW Armory App (If you play WoW. Though its not universal so its 2x mode)
Games:
Carcassone (Board game)
Galaxy on Fire 2 (No need to play the previous game. Great Freelancer type game with pew pew and trading)
Mirror's Edge (completely different gameplay style than the console/pc title but still great)
Need for Speed Shift (IMO its one of the best racers on the iPad up there with Real Racing though I haven't played Real Racing 2)
Pinball HD (Best pinball game on the platform)
Plants vs Zombies (if you haven't played a version of this game, this is the one to get)
Reckless Racing HD (overhead view racing game thats a lot of fun)
Samurai and Samurai 2 (Hack and slash both a lot of fun with neat comic book style story panels)
Time Geeks HD (Amazing 8 bit style Where's Waldo game. I've had a ton of fun with it)
Vector Tanks and Vector Tanks Extreme (Battlezone for your iPad though only Extreme I think is universal atm)