Do I need to set up individual Apple IDs for all my kids' iPads? I will be giving my three children three iPads as their christmas presents.  I am now currently using my iPhone 4 with my only one Apple ID for the App Store.

My question is if I set up their iPads using my Apple ID that I am currently using now, does that mean that if they download a game or anything from the App Store, will it appear on my iPhone 4 as well?  And if I delete the app from my iPhone, will it delete on their iPad too?  If that the case, should I set up a different Apple ID for them?
I strongly recommend you keep a separate ID for each person. When buying a song, for instance, you can share among 5 authorized devices. My computer, iPad, iPod, and iPhone make four. 
Now that my wife and daughter have iPhones, and daughter, a MacBook, I glad they have their own accounts. No crazy apps I don't know about clogging my devices. 
You want your own apple ID and iCloud. Really.

Edit - Using your Apple ID for Apple services will help clarify the iTunes vs iCloud ID usage.