First, welcome to Android! One of the major benefits of this open operating system is flexibility - instead of one-size-fits all, you can choose to replace any app or function with something different and better - for you.

For links, see my Appbrain list of apps for new Androids. If you get an AppBrain account (and install the AppBrain App Market & Fast Web Installer), you can select all these apps and install from the list. And since this list changes as apps improve or new apps are released, you can follow the list.

I've tested over 500 apps - very few of which were games. Since games & social apps are more subjective, my list includes security , replacement of standard apps, widgets / productivity, and utilities. Most are free.  2 that aren't on the list because they aren't on the market: Amazon Appstore and GetJar (both offer free paid apps). It also doesn't include apps that require root access (there is another list for that).

Security: avast! Mobile Security (with Theft Aware), Wave Secure, Carrr Matey

Replacements: Dolphin Browser, Handcent SMS or Go SMS Pro, K-9 Mail, SwiftKey X, Winamp, (Elecont) Weather, Radar, Alerts, Quakes

Productivity: Tasker or AutomateIt (free), Astrid Task / Todo List, Jorte (calendar), Evernote, Google+, GroupMe, Pulse News, Vlingo, Voice Alerts

Interface: Aix Weather Widget, Audio Manager, Auto App Organizer, DayWeekBar English, Executive Assistant, Fancy Widget, Make Your Clock Widget, QuickDesk, SysInfo Widget, WiFi Manager, Widgetsoid

Utilities: CacheCleaner NG, JuiceDefender, DiskUsage, AntTek or Astro or ES File Explorer

Enjoy!