How do I transfer all SIM card details from Nokia Lumia 610 to a new SIM card for iPhone 5S?
It would better that you keep your contacts synchronized in cloud so that you won't have to bother about it being stored in SIM.
Firstly,copy the contacts present in your SIM card to Lumia 610's phone memory -
then follow the steps mentioned in the below link -
Then,you can also download the .csv file from Google to have the backup file with you in case you are out of internet connectivity.