Question: I would like to create a system in which anyone can hook up their email, and have all emails that come to them go through my server first. Anyone who signs up for this service, will have their email vetted by my servers filtration system. If it passes through the filters, I'd like to forward that email onto their inbox.

Answer: 
The vetting part will be quite simple. The difficult part will be ensuring that the user's email passes through your server first. This will either require the users installing some sort of software on their system to intercept the mail coming from their mailboxes, or it will require the users changing their email to a domain you host (maybe a subdomain of their primary domain delegated to you?) or it will require the admins of the mail systems forwarding mail to you when it arrives for your subscribed users.

None of these things are easy, simple to implement, or foolproof.
in the gmail settings you can add forward addresses and delete the original mail in the gmail inbox. forward the mail to a mailaddress on your server. on your server install the filters (spam, virus, content) you want.