Like Jacob said, a simple form of Email Encryption doesn't really exist. However, you can definitely train your users to use S/MIME to encrypt their emails. I find S/MIME to be easier than GPG and is supported by a lot more clients.
To set up S/MIME, you are first going to need your users to get S/MIME certificates. Comodo offers free S/MIME certificates that are quite easy to get. The only downside is that you need all your users to get the S/MIME certs. If some of them don't get the certificates and install them, you'll either end up with unencrypted emails (if sent unencrypted) or unopenable emails (if received encrypted), neither of which are good.
Installation of the S/MIME certificate varies between client, so I recommend looking it for each client that your users use. It's quite easy for OS X Mail, but can be a bit more complicated on something like an iPhone. There isn't much you can do to get around the installation process, but it's more of a trouble for you, than for your users. Your users shouldn't be accessing corporate email on their personal iPhones or devices if the emails are sensitive anyways. Note that if there are different mail programs installed (OS X Mail, and Outlook), the certificates would need to be installed for each client that you want to send and receive encrypted emails.
Now, once certificates are installed on all client machines, the fun starts. Because S/MIME relies on public/private key technology (same as GPG), there is some public key exchanging that needs to happen. The first time the sender (User A) wants to send an encrypted message, he needs to get the public key of the recipient (User B). Now, this is accomplished essentially by having User B send a signed message over to User A. An example exchange is below.
From: User A
To: User B
Hey! I want to send this document to you encrypted. It's the first time I've sent something encrypted to you, so can you reply with a signed message? I've signed this message so your replies to me are also encrypted.
From: User B
To: User A
Hey! I have signed this message and you should now have my public key.
From: User A
To: User B
Great! Here's the document that's been encrypted with my private key.
Now, once the keys are exchanged, there is no need for this exchange again. The keys are stored in the computer's "keychain" and are saved. It's recommended to back up the keychain if important, but it's not necessary. You don't need to exchange keys for every single user at first. You can always have them do it when they need to send encrypted stuff.
That's the basics for sending encrypted emails with S/MIME. It's a little bit complicated but not too much so. The hardest part is just getting the certificates installed on each individual mail client.
EDIT: You should get SSL certificates for your website too, not just your email. You can use Let's Encrypt for free, valid SSL certificates, as long as you have root access to the server. There are also implementations of Let's Encrypt that don't require root access, but I can't personally vouch for those. Some shared hosting providers also support Let's Encrypt, so ask them about it.
