There are ways of doing this but there is not a simple way that will work across all email client / email deployments / Operating Systems.
There are plugins that can talk to Outlook, or work with the back-end Exchange Server, you can write some middleware that will do the same, AppleScript will work on OSX for Apple Mail, ...
Lots of options, but there is nothing pre-canned that will work across all platforms. There is a fair amount of work involved to get there.
Any Idea where I could start my learning process for this?
FileMaker's website has a list of known plugins somewhere. Otherwise, a web search for "FileMaker plugin email" will result in a list you can work through.
Being as 'cross-platform' as you already are (or imagine being in a while) I would advice against doing this on the client side. It can work, but surely depends on the number of users.
First, I would do the analysis: What are your business needs? How big a percentage of outgoing/incoming emails would you like to store in FMP? What is the likelihood of acceptance, i.e. is it realistic to have users send/receive emails in FMP rather than Mail? And many more questions... :-)
I can recommend Mailit 5. I have used it myself in a 10-user-environment. By now, we still only send mail from FMP and receive in Mail, then copy relevant stuff manually into FMP. And it is technically done by running the plug-in on the client computer, although with the automated FMP12-installation process.
So depending on your scripting skills, I would (having done the above analysis) begin by creating a framework of very few, but well tested, scripts. These can be called from various contexts. If your expected user count is around or above 10, consider doing it server-side. But that requires a slightly different approach, needing to have a queue table and a scheduled script running frequently.
In short: I would start with the free trial of Mailit 5 and the demo files they supply. Then try by yourself to write a context-free script and call it with parameters from a person layout, then from a project layout (if your database has such), and so on.
In my solution, we sent out 1.877 emails during October, with around half a gigabyte of attachments.
Please feel free to ask and comment again during your development effort.
Not sure why you wouldn't start (at least) with the Send Mail script function in FM for the sending part. After messing about with plugins and Outlook, I've found the native functionality within Send Mail works well enough for my application.
On the receiving end....hmm... its almost like you need something that would trigger a record update from Mac mail to FM. The only thing that comes to mind that would glue those things together is Applescript....
I'll be interested to see if anyone else has other ideas.
One thing to also consider is whether the iOS platform will ever enter into your solution. In addition to being a FileMaker developer and FBA member, I'm also a Daylite Master Partner. Daylite is a Mac based CRM application developed by Marketcircle. Years ago Marketcircle developed a plugin for Apple Mail called Daylite Mail Integration (or DMI). It was renamed to Daylite Mail Assistant (or DMA) 18 months ago with the release of Daylite 4. DMA allows a user, while still in Mail, to link an email to the history of a person, organization, opportunity, project, appointment, and/or task. Virtually all of the record 'types' in Daylite. It is one of Daylite's strongest features.
In addition to Daylite for the Mac platform, there is also a Daylite for iOS. Probably the number one feature request that I've had from the majority of my Daylite customers is for something similar to DMA on iOS. Many of my Daylite users are using their Macs less and less, and using their iPads more and more. Unfortunately Apple doesn't allow for plugins on the iOS platform. Marketcircle is looking at alternatives but they haven't found an easy to implement, workable solution.
As others have suggested, using the Send Mail script step could the function you are looking for for outgoing mail. Unfortunately it only works to a point. If the user cancels the sending of the email message, you can't change what might have been written to the database. You can't be sure to capture the contents of the email, because the user can modify that after the script step opens up the message in Mail. It's a start, but not what I would be looking for in an ideal solution and it's not what solutions like Salesforce can provide.
For me it remains a function that I feel should be in the core FM product, not in a plugin or third party solution.