My iPad was completely usable the instant I took it out of the box.
I immediately went to the App Store and bought/downloaded a bunch of apps. Had I not had an iTunes library on my computer already, I could have bought music/movies directly on the device - no computer needed.
As other have said, you can have the Apple Store (or other place of purchase) activate it for you, completely negating the need to plug it in to a computer.
The *ONLY* thing that requires a computer right now is upgrading the OS. And you can certainly use it as much as you want without upgrading the OS. (I have a co-worker with an original iPhone that is still running whatever the first iPhone OS that supported apps is.)
It sounds like someone else used your iPad first and set it up for you.
As you said, you "immediately" went to the App Store and bought, which means your iTunes account info had already been registered. Or you're distorting reality here and simply forgot about the initial setup.
Do you recall setting up your account in settings?