As has been mentioned above, Aegis is a good choice for Android. It is entirely open source, keeps all your data encrypted locally, and allows you to perform encrypted back ups and exports so you can save your database on an external device for easy recovery should your phone be lost or damaged.

For Apple, I believe the best option is Tofu authenticator instead. It is also open source, but unfortunately does not allow local back ups, instead opting to back up your database encrypted inside your iCloud Keychain if you choose.

Whichever option you choose, you should still write down the shared secret for each site on paper when you first enable/switch over your 2FA.