Calvetica and
Agenda are both significant improvements over the built-in calendar, and they run off the same data, so no worries about syncing.
NAVIGON is the best GPS app, but is quite spendy.
If you like podcasts, you
need to check out
Instacast.
For reading purchased eBooks,
iBooks and
Kindle are both very good. For reading RSS feeds
Reeder is the best Google Reader client. And for reading longish articles that you came across via your desktop computer's web browser, the old favorite is
Instapaper and the newcomer is
Readability.
Due is a wonderful reminder app whose primary trick is that it will keep nudging you until you actually do whatever task, or admit that you're rescheduling it for later (think, "to-do's with snoozing"), while at the other end of the scale,
OmniFocus is an awesome, hardcore, to-do app in the
GTD vein; it has iOS and
Mac clients, with syncing -- they're
very spendy, but worth every penny if your mind works that way.
For an office suite, you could do much worse than Apple's
Pages,
Numbers, and
Keynote.
RedLaser scans barcodes and tells you what they mean.
If you travel much (or your SO does),
Flight Update Pro is an excellent way to keep track of flights.
There are a
lot of good photography apps for iOS (if you're curious, check out the
LifeInLoFi and
iPhoneography websites), but you can get pretty far with the built-in camera app and
iPhoto.
Tweetbot is my favorite of the Twitter clients.
Day One is very good for keeping a journal/diary, syncs to DropBox, and there's a Mac client available too.
The
Yelp client is quite good - crowd-sourced recommendations for local restaurants/shops/etc. become all the more useful when the thing you're looking at knows where you are right now.
If you get a lot of packages via UPS/FedEx/etc., then
Delivery Status is pretty cool for tracking them (and there's a nice dashboard widget for the Mac).
1Password and
DropBox should be on everyone's computers and iOS devices -- 1Password stores all your other passwords in an extremely secure database, protected by... 1 password; so if you can remember that one, it takes care of everything else for you. And DropBox does a lovely transparent job of syncing one folder between your Macs, PCs, and iOS devices. Lots of other software on iOS makes use of DropBox for syncing and backup (including 1Password).
If you subscribe to Netflix streaming, you
really want the
Netflix app, which will let you watch things anywhere; and if you pay attention to movies ("who was that actor in that movie with that actress who was in that other movie?"), the
IMDb is very handy.
If you're the least bit interested in astronomy,
Star Walk and
Solar Walk are excellent.
As far as games go... If you're a fan of the newer style of strategy board games (aka eurogames), there are a lot of fabulous choices, like
Carcassonne,
Tigris & Euphrates,
Puerto Rico,
Caylus, and
Ticket to Ride. If you like chess, try
Shredder Chess. I've tried a lot of tower defense games, and I still keep coming back to
Sentinel 3. If FPS's are your thing, try
N.O.V.A. 2 and
Rainbow Six. For "bullet hell" shoot-em-ups (think, the great-great-grandchild of Space Invaders, on steroids), anything by Cave is great, but I particularly like
Espgaluda II.
Zombie Gunship won't show off your hardware very well, since it spends most of its time emulating a night vision display, but it's pretty addictive/immersive (simple plot: you're in a gunship, shoot the zombies). On the other hand, both
Infinity Blade (and its sequels) and
Real Racing 2 HD will look gorgeous on your new iPad, and are a lot of fun to play.
And, finally, for finding other apps,
AppShopper is
essential, and you might find
AppStart useful too.