You can request a copy of all of your Family App data directly from the app.
Go to More → Profile → Request all your data and follow the prompts to choose a password that will be used to protect your exported information. Your data will be prepared in a password-protected ZIP file, and you will receive an email with a link to download it. The export includes your data in JSON text files along with any photos you have uploaded, which are provided as JPEG files. Preparing the export may take some time, although it is often available within about 30 minutes.