I want to forward Apple mail to G mail account I am running OS X Mavericks.

I want to forward my apple mail (happens to be verizon.net) to my gmail account.

I want to read all mail in gmail. 

Is this possible?
Apple mail is just a mail client, to forward your mail to gmail, the set up needs to be done on Verizon. 

You will need to log into Verizon web mail and then set up forwarding from there to gmail, otherwise you are forwarding your mail twice. Once from Verizon to your computer running Apple Mail and then again from your computer to google.  Depending on how good the Verizon web mail/mail server is forwarding should be easy to set up.

What you want to do is cut out your computer all together.



If you want to really set up forwarding from your computer i.e. from Apple Mail you would set up a rule in the following way:
While in Mail: 

Press ⌘+, to bring up the preferences window

Select the Rules tab



Click "Add Rule"



Put in your "Description"

Select "Every message"



Under "Perform the following actions" 

Select "Redirect Message" to put in your gmail account.

Please note that these instructions are for Mail 5.3 but should work with newer versions.