Lot of apps were shared in this thread:
New iPad (iPad 3)
What do I do on my iPad?
Well lately I'm spending lot of time glued to the desktop PC, so my iPad spends more time in a dock. It's sitting to my left, with a bluetooth keyboard. I often have the email app, or a RSS reader, and when I tap the space bar, it wakes up instantly and refreshes the feeds.
When I'm using the PC, I also often listen to music with Track 8.
I sometimes just turn around and check new threads with Alien Blue, a Reddit client, where I follow just a few subreddits of interest.
If I just got up and want to note something and the computer is not on, I might simply send myself an email.
I get notifications as well popping up, from the Google calendar, which is very useful.
I like Google Drive as well, I started using Google Spreadsheets more often lately to take notes. I mark the documents as "offline" in Google Drive app, then I can look them up on the iPad.
Basically, when I'm using the PC it's a second screen with some very useful functionality that removes the need for a lot of alt tabbing.
Get something like AppShopper then favorite the apps you like, and you get a notification now and then when price drops or it goes free.
Oh yeah Podcasts obviously... except it's not that obvious, as you don't get the iBooks and Podcasts apps from Apple installed by default.
I also use Twitter app to follow some streams that don't have RSS.
If you let other people use your iPad and you want a more private space for your favorite photos, PDFs, videos, text files ... things you don't want your colleagues or family or children to mess around, I quite like st@sh. It's very well made.