Changing Apple ID to alternate email address

I'm not sure what you are thinking about doing. Do you have two Apple IDs. If yes, delete away, but you might just want to deactivate it.

See How to deactivate your Apple ID account  https://support.apple.com/en-us/HT208503

If you request account deactivation, you'll receive a unique alphanumeric access code. When you're ready to reactivate your Apple ID, contact Apple Support, provide your access code, and we’ll help you reactivate your account. If you lose this code, we won't be able to help reactivate your account." 

See How to Delete your Apple ID Account https://support.apple.com/en-us/HT208504 

“While your request is being processed, you can contact Apple Support with the unique alphanumeric access code you received, and we’ll help you cancel your request. After your account is deleted, Apple can't cancel the request, reopen your account, or restore your data.”

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But maybe you are thinking about using an alias for you Apple ID and deleting your Apple ID. You don't want to do that. See Change your Apple ID  https://support.apple.com/en-us/HT202667, this section How to change your Apple ID if it ends with @icloud.com, @me.com, or @mac.com.

"If you use an @icloud.com, @mac.com, or @me.com email address as your Apple ID, you can’t change it to a third party email address. But, you can change it to an alias of your Apple ID address."

So, you could do so if that is what your mean by "I want to use my alternate email as my Apple ID and delete the original email I used. Note that if this is what you are talking about, there is not much reason to make the change. You can just use the alias; it can do everything your actual Apple ID can do.

If you decide to change your Apple ID, see:

Change your Apple ID (Be very aware of the restrictions here.) 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 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

Apple hasn’t published any guidance for using an address again. Normally, if your e-mail address was associated with an ID in any way, for example as a Reachable at or Rescue e-mail address, then it can't be reused with another Apple ID for approximately one year. I have also seen posts that stated Apple told them there was a 30 day waiting period before the address could be used again.  You will need to get another e-mail address to use as your Apple ID or you can try waiting 30 days to see if that works.

Note the restrictions in the article.

Apple ID Name Change.        

 What to do after you change your Apple ID or password - Apple Support    I suggest doing this before you change your Apple ID or password.