A bunch of random stuff:
The Elements book/app is the first thing I bought for my original iPad. It’s a guide to all the elements of the periodic table with little essays about each one, physical properties data provided by Wolfram Alpha, and beautiful 3-D photos of samples of, and artifacts made with, each of them. The photos can be spun around with your fingers. Also, Tom Lehrer’s Elements song is built in. It’s basically a coffee-table book with multi-media. Nicely done.
Threes is a horribly addictive game. Simple and elegant. It costs a couple of bucks, but my amortized cost per hour of entertainment is probably down to about a nickel by now.
If you subscribe to HBO, the HBO GO app is great for watching shows on demand from anywhere. Same for Netflix. Really, if you are a customer of almost any business, they probably have an app.
Favorite weather apps are Dark Sky and Check the Weather (which uses Dark Sky data). Clean interface and presentation of data, and fantastic for ultra-short term prediction (e.g. “Rain will start in 7 minutes and continue for 30 minutes.”) Both paid apps, but worth the few bucks.
If you want to use the iPad as a musical instrument, your choices are quite broad. You can do a lot of stuff with Garage Band, which may be free with new iPads now(?). Animoog gives you a skeumorphic Moog synthesizer to play with, complete with rheostat dials and sliders. Bebot is another good synth music instrument–you can play along with songs from your music library, I think. Musyc is kind of a rhythm toy. You drag around barriers and shapes that bounce off them (driven by simulated gravity) each shape type makes different sounds, so you can build up very complex rhythmic patterns. Very fun.
Waterlogue is a very cool app that lets you turn photos into simulated watercolor paintings. For basic photo editing I think there’s a free Photoshop app, in addition to Apple’s own iPhoto. Brushes is a really nice app for creating your own “paintings”, either on a blank canvas or as a layer over a photo from your library.
If you have not already, get Find My iPhone, and also enable that on the iPad. You can log in on any browser and locate the device. This is great if it gets stolen, or even if you forgot which room you left it in, since you can have it “ping” audibly, even if the volume’s been muted. On a related subject, be sure to put a passcode lock on the device, since it will make it impossible for thieves to wipe the device and fence it before you can locate it with the Find My iPhone service.
There’s no reason not to download free apps just to try them out. You can always junk 'em if they turn out to be crap. On the other hand, I have found that my most-used apps are ones I paid a couple of bucks for. Mostly, that’s because they tend to be ad-free and more carefully designed and coded.