I don't think you need to use a Virtual private server at this time unless you are getting so much traffic that GoDaddy is limiting you.

Here is info from Wordpress about email forwarding:

https://en.support.wordpress.com/email-forwarding/

Seeing as you probably bought your domain, I really suggest a Google Apps for Business account, because even though it is Gmail, your emails will come to whomever you send to as from your domain. There are a lot of other features with it as well, you can look at it from here:

https://apps.google.com/intx/en_us/products/?utm_source=google&utm_medium=cpc&utm_campaign=na-us-en-apps-bkws-gafw-trial-e&utm_term=google%20apps%20for%20work&KWID=43700007880944900

If you need any help, I can for sure help you with your email, as well as work out how to get Wordpress to do what you want. Even if you don't go with Google Apps, there are lots of other ways to get done what you need.

For sure though, if the current emails are being marked as spam for your users, or not even getting delivered that is not good for your reputation.