I'm not very familiar with TestFlight. I would like to add a build with a new version for my app. Do I need to add the new version to the App Store settings first?
For every build you upload to iTunes Connect you can decide whether you want to share it with testers on TestFlight and/or submit it for app store review.
When you share builds with TestFlight Apple will review these builds too and you can receive warnings to them too. The initial TestFlight build will be tested more extensively.
Do I need to add the new version to the App Store settings first?
I don't know about which app store settings you are talking about, but all you have to do before uploading to iTunes connect is set the code signing and provisioning profiles first in XCode and Apple's development portal and make sure you increment the build number on every new upload because the upload will fail if you try to upload a build with a build number already present on iTunes Connect.
Once uploaded iTunes Connect will start processing it and then you can share it for testing with TestFlight or submit for app store release review.
You can read more info about TestFlight in Apple's dev site