Congrats on being a new iPad owner!
Yes, that's actually how the iPad's operating system works. When you press the "home" button (the round button with a square symbol) to exit an app to go back to your home screen, it doesn't completely close down the app. It goes into a state of suspended animation, ready to be resumed where you left off when you come back to the app. Now keep in mind that the apps you exit out off DO NOT run in the background (so they don't constantly consume memory or power). They're just in a "paused" state ready to resume when you come back. The exception to this are apps that can play audio in the background, apps that require downloading in the background (digital magazines, podcasts), GPS location services, mail, and some accessories.
So for example, you can be posting via the app for this forum and need to look up something on Safari. You hit the home button to leave the app in the middle of your post, open safari and do your searching, come back to the app and you'll be exactly where you left off, mid post. Now if you go back to Safari, you'll be on the same page you did your search on. You can then watch a YouTube video, pause it to edit an excel file, then reply to an email, and go back to the YouTube vid you paused earlier, or the excel file you're working on editing.
When you hit the home button twice, it'll bring up the task bar. This show you what apps you've accessed recently and are on standby. You can use this as a shortcut rather than finding the app on your home screen. Now if you want to completely close out of an app so that it starts fresh the next time you open it, you'll have do do it here on the task bar. Just hold down on an app icon until they "jiggle" and press the "X" of the corner of the app you'd like to completely close.
Here's a neat trick, the 4 finger swipe. While in an app, Place 4 fingers on the screen at the same time and swipe to the right. You'll immediately jump to the last accessed app. Swipe right again and you'll be at another app you used prior. Swipe left to go back to the previous app. It's a handy shortcut if you don't want to press the home button to go back the the home screen or task bar. It sorta kinda works like alt-tab on a pc.
Enjoy the iPad! I hope it'll be a great experience for you.