Should I activate my phone first or transfer the data?

You cannot transfer data from the old iPhone if your new iPhone is not activated.

Best you make a local backup of your old iPhone using computer iTunes then restore the backup with your new iPhone.

Read here: Transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support

newSVD Said:

”which should I do first to my new unclocked iPhone: go to my carrier to activate it or transfer the data from the old phone?”

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That’s your call. But, In whatever way possible (iCloud, perhaps), I’d most certainly consider transferring the data from the old phone, that way no personal data is shared.