I'm running the most recent software update on my Mac however, I'm getting a message on Safari saying my browser is out of date. How to I fix this issue?

I think I can explain what is happening. If you are unable to access certain web sites and possibly getting a message that your browser must be updated, this is not an unusual situation. The reason is that web site engineers write their software to work with various browsers based on standardized specifications if they are to be compliant. Some sites are not yet compliant. Also, some older browsers may not be fully compliant. You can hope that your web site's developers eventually update their site to work with Safari, you can upgrade Safari by upgrading your OS or you can use another browser that does work such as Firefox or Chrome. They can be individually updated as needed. It is important to know that Safari is installed as part of the operating system and it normally it cannot be individually updated. Safari is periodically updated when the computer's operating system (OS) is updated or upgraded. Finally, even if you are able to update Safari, there is no guarantee that it will be compatible with every web site. This is why many of us have more than one browser installed.