1. Normally, you don't every turn it fully off, you just to put it to "sleep." Click the top button, and the screen goes dark. The dark screen is how you know it's asleep. Sleep is a low-power mode that can last for many hours and still receive calls and check email (if you've set automatic email-checking). This is the same as closing a flip-phone: it's still "on" for taking calls, but is in power-saving mode.
If you want to turn it completely off so it's non-functional (to store it without using any battery power at all), you hold the sleep button until you see the red slider. Slide that, and it will power down. But that's not something you would normally do.
2. You don't need to worry about managing apps. It's all automatic. (Typical background programs are the phone app, the iTunes player app, the mail app, and the browser. Third-party apps/games do not run in the background, although their push notifications keep on working anyway--so you can get AIMs for instance, even when AIM is not really running.)
If an app starts crashing, it's possible that RAM is being hogged by something, but the simplest answer is just to power your phone down completely and turn it back on. This is rare, but some games do run out of RAM sometimes.
Enjoy
