I don't want to sync all my contacts with my Google account. I would like to just keep my contacts on my phone and back it up myself. I have disabled contacts syncing in the accounts settings on my phone, but when I try to add a new contact through the 'Contacts' app, it tells me that my new contact will be synchronized with my google account.
When adding contacts to the phone via Google's built-in Contacts app, it will give you a choice how to save the contact. While adding a new contact, tap the "Google contact" to expand the drop-down for more choices on where to save it, and choose the option to store it only locally:
Keep in mind that various phone manufacturers such as Samsung, HTC, Motorola, etc. load their own versions of some system apps, including the "Contacts" app. Your Contacts app may look differently, and have this option elsewhere (or may be completely missing it.)
You can go to
App Drawer > Select Phone > Tap on the three dots > Select "Import/Export" > And then select "Export to storage". This will back up all your contacts on your phone itself. You can later on import from this backup if needed. However, you need to do this manually every time you modify your contact list.