Hello all, I know I'm new here, but just thought I'd share my 0.02 cents worth. When I originally got my S3, like all of you I was a little shocked (coming from an iPhone 4 that lasted 3 days without needing charged), however I had to keep reminding myself that these are not your standard flip phones anymore. What you're asking these phones to do, is nothing more than what you'd ask your Laptop to do. Yes, as soon as you put it to sleep, it's going to conserve battery power, but think of it like this, you play high end games, or run a high end application (and on these phones high end applications can be word processing, email, web browsing), then yes, the drain is going to be more and more.
What I did notice prior to updating from Ice cream sandwich to JB is this, the more I used the phone, the smarter it got. Make sure you hold down the home key, and kill all apps you're not using, clear your cache out by holding down the home key and checking your task manager, and usually you will find that your phone can run about 2 days with a full charge. So far (knock on wood) I've had 2 days straight before a charge is required.
As for the wi-fi, here's a simple solution to resolve battery issues. Not sure if you all know this or not, but by default the wi-fi is not set up to sleep when the phone is set up to sleep, so do this...
Go to Apps, Settings, Wi-Fi (Tap on Wi-Fi, not the toggle switch), Tap on the Menu button in the bottom left corner, and select Advanced. There will be an option that says Keep Wi-Fi on during sleep. Change this to never. This will put the Wi-Fi to sleep when the phone sleeps, thus saving you a ton of battery time. As soon as you wake the phone, it automatically reconnects to the Wi-Fi source and you are good to go.
Hope this helps.
Orion