My friend said when using an app, pressing the home button saves battery life instead of pressing the back button. He thought by pressing the home button, the app stays in RAM so that when you reopen it it just open, android doesn't need to create the app and store it in RAM, which costs battery. Is it true?
That's basically what Android does in many cases, yes. But I don't think it's true that it will have all that much of an effect on battery life. Most apps don't take long to start up at all — it would take repeatedly opening apps that take multiple minutes to start to noticeably affect your battery life, I would think. Android does automatically close apps when needed, and apps normally can't run in the background (just services), so there is not much reason to back out of an app rather than just using the home button.
When you click the Home button, it keeps the app in background, which means it's running. When you click the back button, it kills the app (but not always; it depends on how the developer implemented the app).
I'm not really sure about it's effects on the battery, but the more services / apps you have running in background, the more battery it eats up. It's running after all. And if you're dealing with games, I would suggest exiting it properly (the back button will usually ask you if you want to exit the program; again, it depends on how the developer implemented it).