iPhones don't store contacts or any data in the sim card.
The sim card only holds your number and the settings for your cell service.
If the iPhone displays cell service at the top right corner, then the sim card was correctly identified.
If it displays a "No Sim" message , then it means the sim card has an issue and was not detected.
For your contacts, if they were stored locally on the iPhone, then restoring a backup should have brought them back. If they were stored in iCloud or any other service such as Gmail or Hotmail, then adding that account in Settings->Passwords & Accounts would bring them back.