You could do that by choosing "Set up manually" in step 2 and "Don't transfer Apps & Data" in step 5 of Set up your iPhone, iPad, or iPod touch - Apple Support
Then use iCloud to transfer Contacts and Notes by turning on Settings > [username] > iCloud > Contacts and Settings > [username] > iCloud > Notes on the old phone (if not already on) and turn on the same options in the new phone.
This assumes you are using the same Apple ID for iCloud on the old phone and the new phone.
The following is unrelated but may be of use for the old phone after you no longer need it: What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch - Apple Support