iOS does allow importing of contacts from a SIM card (Settings > Mail, Contacts, Calendars > Import SIM Contacts), but iOS does not store contacts on a SIM card, so if you upgrade from one iPhone to another, there is nothing to import from the SIM card, because there is nothing stored on it to begin with. The correct way to transfer data from one iPhone to another is via iTunes backup/restore.
But you can follow this step by step guide about How to transfer SMS and Contacts from iPhone to PC, including all files such as SMS,Contacts,Music,Photo and video etc.
For Mac:How to transfer iPhone files to Mac
For PC:How to transfer SMS and Contacts from iPhone to PC
It's easy to understand and works pretty well for my iPhone.