While you can also create an iCloud backup of your device, I've found that backing up to iTunes is best since it's faster, more secure, and you're not bound by the "free" iCloud storage size limitations.
Plug your old phone into your computer, open iTunes, and click the small "Device" button.
You will now see the settings pane for your iPhone. Check the "Encrypt iPhone Backup" option. This option will prompt you to create a password to unlock the backup. Additionally, if you're transferring Health and Wellness data from your iPhone, you must encrypt the backup in order for this data to transfer over.
Finally, click "Back Up Now".
Safely eject your old iPhone from iTunes and power it down. Using a thin paperclip, remove the SIM card tray on the side of your old iPhone and insert the SIM card into into your new iPhone.
Connect your new iPhone to your computer and turn it on. You should see a white "Hello" screen. If you do not see this screen, it means your device has already been set up and you first need to wipe it by navigating to Settings > General > Reset > Erase All Content and Settings.
Once you see the "Hello" screen, open iTunes and click Restore iPhone. Select the backup you just created, and click Continue.
The restore should take about 5 minutes; afterwards, your iPhone will automatically restart. Finally, safely eject your new iPhone from iTunes.
You're almost done! Now that everything has been transferred over, I recommend securely erasing your old device so that it can be resold or sent back to your carrier (if this is a replacement or upgrade device). I wrote a separate guide on securely erasing your old iPhone.
Most iPhone users use their iPhones everyday without accessing all of its capabilities.