I am now testing 2 plugins. But I had to use a SMTP plugin and add DKIM and SPF records.
One thing I have learnt is not to use the same email address to send and receive the email.
As a learning curve I have found a lesson to show how to make a plugin. I have got this working, it displays the form and sends the email. I am now working on how to stop it sending blank emails. Also adding security check such as 1 + 1 =. I am using wp_mail as mentioned by @sneakyimp.
What is the best PHP Editor for Windows?