Using 2 factor authentication is one of the most important things you can do to protect your digital identity and it is simple to do. 2 factor authentication means that your password and one other form of authorisation is needed to use your account. This is usually done by text message or email (which could be less secure) although authentication apps, such as google authenticator, are increasingly being used and can provide even greater protection.
Most major service providers including social media will offer 2 factor authentication and my recommendation is that you enable it for all of your important accounts. Banks and government agencies should require this by default. Enabling it for all of your other accounts means that you will have an extra layer of protection if your passwords are compromised or something else goes wrong.
There is a risk that you will not be able to authenticate if you lose the means to do so. This could happen if you lose your mobile phone or change the SIM when switching providers. For that reason you should always provide an alternative contact or email and setup the backup approach which might be secret questions or back-up pass-codes that you can print and save.
This is another reason why it is important to keep your details up to date for your important accounts and definitely do this before and after changing your phone number.