Power saving mode (by default) turns off some vibration, but it should only turn off the "haptic feedback" vibration (vibration when you touch certain UI controls). You can change that by going to Settings, Power saving mode, and unchecking "Turn off haptic feedback". That's the mode I run my SGS3 in all the time.
To get vibration working for text messages, you also need to enable vibration in the settings of the Messaging app itself (I'm not sure which settings you were referring to when you said that you "went into settings and enabled everything that has the word "Vibrate" in it"). To do that, go to the conversation list in the Messaging app (not an individual conversation), hit the menu button, then select Settings. Look for the "Notification Settings" section and check "Vibrate".
To get vibration working for incoming calls, there is a similar setting in the Phone app. Launch the Phone app, make sure you're on the Keypad, Logs, or Favorites tab, tap the menu button, select Call settings. Then select Ringtones and keypad tones and select Incoming call vibration. The Device vibration will let you modify the vibration pattern used (so you can tell an incoming call from a text message / email).
To get to the Phone and Messaging apps: Press the "home" button at the bottom of the phone (the physical button in the middle). That brings you to your home screen. The touch the "Apps" button (it looks like 9 square white dots, and is usually on the lower right of the home screen). That will bring up a list of all applications installed on the phone. Scroll through looking for an app named Messaging (or Phone). The icon for the Messaging app looks like a white envelope with a yellow piece of paper coming out of it. The icon for the Phone app looks like a green square with a white phone receiver on it. Tap the app icon, then follow the instructions for each app above to activate vibration for text messaging and incoming calls.
