There are thousands of Android apps to download and play with. Here are a few suggestions to get you started with your new Android smartphone or tablet.
I posted 9 Essential Apps for New Android Users and Top 5 Business Apps for Almost All Smartphones two years ago, but some of these apps are no longer available or may be replaced by better alternatives. Of the ones I still use, I recommend Evernote for note-taking, Documents to Go for editing Microsoft Office files, Skype for free video calling and instant messaging, Wifi Analyzer to help you improve your wireless network, and PdaNet for tethering.
I also think you should install avast! for antivirus/complete mobile security (free and highly rated by AV-Test, GasBuddy (because we all could stand to save on gas), and Camera ZOOM FX (probably the best camera app for Android).
If you use your phone or tablet to catch up on news and websites, Google Reader, Zite, and Pocket are awesome.
You'll find all of these apps and a whole lot more in the Google Play app (formerly known as Google Market). Pro tip: You can remotely install apps to your phone or device from your laptop or desktop computer from the Google Play website.