Export your contacts as a .cvs or .vcf file. Then import that into your Google (Gmail) contacts online. When you register your Google account on the new phone the contacts will automatically sync (download) to the new phone. Contacts really should be saved to your Google account for automatic backup and easy restoration or transfer to a new device. It's foolproof and super simple.
The option to save contacts to SIM cards is disappearing and no doubt will be gone completely in a year or two. Several manufacturers have already gotten together to eliminate the SIM card altogether. Unlike other contacts options, SIM contacts can have names and numbers only - no email addresses, physical addresses, photos, birthdays, notes - none of the extra information most people like to be able to add. In short, SIM contacts is old, outdated technology that should not be considered a viable option.