Hello emilywatsonx.
Thanks for reaching out to Apple Support Communities. We see that your contacts are not saving. Contacts are very important, and we will be glad to assist.
To save and sync contacts, you can use iCloud or a third-party account, such as your email account.
Use other contact accounts on iPhone
At Settings > Contacts > Default Account > What is shown there?
After checking that, go to the website for that provider and see if your contacts are shown there.
For example, if iCloud is shown, go to the iCloud website > Contacts and verify if missing contacts are shown there.
You'll also want to verify that you have Contacts turned on for the accounts in which you'd like to be using, as shown here:
Manage and delete contacts on your iPhone, iPad, or iPod touch
Once you've verified the settings are all correct, then add a new contact and then verify that it saved.
Add and use contact information on iPhone
Cheers.