I surrendered our first iPad to my wife and daughter but did buy the new version of it -still going about adding the apps I used to use. I'm not including some of the iPhone apps that I also have on my iPad but rarely use.
Caveat - I do not mind paying for apps I like. There are some here that cost me money, but most, I believe, are free. The ones that cost money, Pages and Instapaper, for example, are well-worth whatever they cost.
Here goes...
Entertainment -
ABC Player - ABC television shows, free o' charge. It's basically how I watch Modern Family.
Netflix - Self-explanatory
Vimeo - Basically, YouTube for people with taste.
Trailers - An Apple app that shows movie trailers. I think I like the trailers more than I like most movies.
Crackle - Free movies. Some of them are pretty good - Seinfeld episodes as well. Kind of hit or miss, but free is free.
Vevo HD - Music videos. Didn't think I would use it, but I do occasionally.
HBO Go - Great if you're an HBO subscriber and your internet provider is with the service.
Books -
iBooks - I love the way Apple put this app together. You'll marvel at how turning a page looks.
Kindle - Not as nice to look at or use, but if you buy books via Amazon, perfectly acceptable. I prefer Kindle
Social -
Facebook - If you use it.
Twitter - I like the official app (free).
Tweetbot - Seems to be the most popular third-party Twitter client.
Tapatalk - I do like it, even for the iPad.
Work -
Pages - Great word processing app from Apple. I use it with an Apple bluetooth keyboard.
Keynote - Build presentations, web-pages, whatever.
Dropbox - Free storage and document syncing. Great.
Box - Free storage.
Evernote - A 'junk drawer' for notes, lists, whatever.
Numbers my add on -, also, iPhoto and iMovie
Lifestyle-type apps -
TripIt - Travel organizer. Get an email confirmation of a hotel, flight, whatever reservation, forward it to TripIt and it builds and keeps you itinerary.
Yelp - Recommendations for restaurants, hotels, etc.
The Weather Channel - It's, well, the Weather Channel.
Epicurious - Recipes
Flixster - Movies trailers, reviews, showtimes.
News -
I use AOL's news collector for iPad - Editions
Flipboard - Great. Just get it.
Zite - Great. Just get it. Both are personalized 'magazines' you organize around your interests.
Reeder - Syncs with Google Reader if that is how you organize any RSS feeds you subscribe to, but much prettier.
StumbleUpon - Kind of a random article generator that is geared to your interests - hit the Stumble button, get an article on the topics you follow.
Instapaper - I love Instapaper. It's an offline reader of sorts - what it does is save the articles you want to read for later. It installs via a browser button...if you're using the browser and you come across something you want to read, but not right now, hit the Instapaper button and it will be saved for you.
Where it gets really handy for me...it works with StumbleUpon. I click through the StumbleUpon suggestions and just save the ones I like to Instapaper. (Just a button press). I can get a pretty good pile of stuff to read offline that way.