Luckily the process of transferring contacts and other data between Android and iPhone devices is a lot simpler than it used to be, so this won't be too hard

On your SIII:

1. Connect to a wi-fi network or data if you still have service on the phone.
2. Make sure your contacts & data are synced and up to date with Google by going to Menu > Settings > Accounts > Google > Sync

On your iPhone:
For Contacts:

1. Go to the Settings app on your device.
2. Select Mail, Contacts, Calendars.
3. Select Add Account.
4. Select Google.
5. Fill out your account information in the following fields:
   
   • Name: Your name
   • User Name: Your full Google Account or Google Apps email address.
   • Password: Your Google Account or Google Apps password. (If you’ve enabled 2 Step verification, you’ll need to generate and enter an application specific password.)
   • Description: Description of the account (e.g. Personal Contacts).
   
   
6. Select Next at the top of your screen
7. Make sure that the "Contacts" option is turned ON. The switch should be green.
8. Select Save at the top of your screen.
9. Open the contacts app and syncing of contacts will begin

For your Google Calendar:

1. Go to your home screen
2. Go to Settings > Mail, Contacts, Calendars > Add Account > Google.
3. Enter your Name, Gmail address, and Password.
   
   • If you use 2-step verification, please use an application-specific password instead of your regular password.
   
   
4. Next.
5. Emails, contacts, calendar events, and reminders you send or create on your device will now sync directly to your Google Account on the web. To sync only your calendar, turn the others Off.

For Gmail:
Download the gmail app for iOS here: http://itunes.apple.com/us/app/gmail/id422689480?mt=8