You can use the @me.com now to do anything you can do with .mac.com. The former is an alias for the latter. But if you want to make the change, see the instructions below.
Change your Apple ID (see the part that covers changing @mac.com to @me.com under this heading How to change your Apple ID if it ends with @icloud.com, @me.com, or @mac.com) https://support.apple.com/en-us/HT202667
Before changing your Apple ID using a computer, sign out of all Apple services that use your Apple ID. If you are going to use an iPhone, iPad, or iPod, then stay signed in with your Apple ID on only that device. This instruction is taken from the above document (https://support.apple.com/en-us/HT202667) under the heading Before you change your Apple ID.
For sign out purposes, see Where can I use my Apple ID? https://support.apple.com/en-us/HT202659 “This list is provided for informational purposes only and is subject to change without notice. It's not intended to include every feature or service that uses Apple ID. Visit the Apple ID support page for more information.”
There may be some redundancy, but see What to do after you change your Apple ID or password
https://support.apple.com/en-us/HT204071