How do I get my contacts off of my Apple iPhone5s phone and transfer them to my Apple6s phone ?

ITUNES ...BLUETOOTH....

If you have Verizon as a carrier download the app Backup Assist, Open App, follow the instructions on setting up, then this will back up your contact to the Verizon server. So if you damage or lose your phone your contact will always be available BUT also remember to do the backup periodically...day,week,monthly etc depending on how often you create new contacts

Log into iCloud account on your old iPhone, back up your contacts. Go to your new iPhone, log into the same Apple ID, turn off Contacts in iCloud, and turn it on again, choose Merge. How to Transfer Contacts from iPhone to iPhone