Below are some of my personal App recommendations, Most are free that have more functions in full paid versions. These Apps will really bring out the functionality in your new Android.
SMS App: Handcent: Hands down the Best most feature rich SMS app
Media Player: there are many I like
Meridian & Mixzing they support swipe to advance tracks. meridian seems to be more feature rich.
Other Media Apps: Pandora,
Mvideo player,
iHeartradio,
Doggcatcher podcatcher,
Google Listen podcatcher.
Seesmic: feature rich free twitter client
Good games:
Angry Birds.. Fling vengeful birds at greedy swine
The Moron test buy .99 full version well worth it, keep you busy and get lots of laughs.
Talking Tom: repeats what you say in a funny voice, Much like iPhones "Talking Karl"
Bonsai Blast
paper toss
Reversi othello like game, logic and thinking..
Other non Game fun Apps:
The Swartz; Star wars light sabers...
Google SkyMap: point it up at the sky and it maps out stars and constellations.. my kids love this one.
Kids paint finger paint on the phone and new works of art as .jpeg to share.
App Killer:
stock task killer As Some have already mentioned (Long press on Home key)
Camera App:
Picsay my favorite photo editing apps
Productivity Utilities:
RingDroid: Ringtone app create and edit your own ringtones
Smart Measure: Measures distance bubble, height..
ConvertPad: Unit converter
Google talk: should already be installed
Google Shopper ,
Shop Savvy: can do research by scanning Barcodes, also enables QR code reading.
Barcode scanner QR code reader and barcode scanner
Wikitude: Augmented reality App.
Layard: another augmented reality App
DropBox: view files from any remote PC on your mobile
Quadrant standard: speed and graphics test on your phone
Shootme: enables screen shot capture
Connect Bot: Terminal emulator
Astro file manager is nice.. but i find stock file manager works just as good.
And one of the best for last.
Launcher pro replaces stock TouchWiz. I downloaded this so I could change the default docked icons.
My advice on getting started is to read the help sections, & tips & tricks sections of the Android forums
Help & How To
Android Apps