"What mechanical/manufacturing skill (or skills) have you personally found most rewarding, and why?"
That would be hard to say, the overall ability to manufacture (in the most basic meaning of the word) is the most rewarding. I can manipulate matter into things, what I imagine I can make real (within limits, unfortunately). That's the profound part of it.
As for a specific skill that I am most proud, or find most useful, I would have to say that a short list would be: Filing (Using files, Nicholson has a little pamphlet titled "Filosophy"...), Turning (using a metalworking lathe), and probably drafting (drawing, and most frequently for me, CAD, or computer aided design), although it is not directly productive.
I tend to collect skills - or rather I collect the ability to perform processes, so I do small projects where the end is not nearly as important as the process. My latest success is detailed here:
"Or to put it in other terms, I'd be interested in getting some sort of glimpse of what a life of mechanical philosophy (it does seem the best phrase for it, doesn't it) is like from the inside."
Well I'll spare you the book that I will never but ought to write. I am of a small subset in the field as I am both amateur and professional, although I have done jobs for pay, I have equally done jobs for pleasure. I sell small machine tools through the internet, which allows me to interact with both professionals and amateurs, so I have a sense of how diverse the field is and I get to communicate with many interesting people (and quite a few nuts)...
There are two spheres of the mechanical/manufacturing world, the amateur and the professional.
The amateur worlds are many, all other human activities may be aided by mechanical acts either to make tools, instruments or objects that further that activity (for instance the amateur shooter learns gunsmithing, the biker learns engine machining) Add to that the world of the amateur machinist, or "Home Shop Machinist", who may make tools to make more tools, as well as for the various hobbies.
The professional is much like the amateur world except that they joy of business comes into the practice. As well some professionals consider their trade only as a job, and take no joy in it, when an amateur always (by definition) does.
And both worlds cross over, as there are professionals who are also amateurs...
Clear as mud all this, but because we have the wonder of the internet I can give you some leads...
"Another possible question: what would you recommend for someone just starting out their exploration of mechanical/manufacturing issues?"
For professional manufacturing:
Modern Machine Shop Magazine
Machine Design Magazine
If you get the Travel Channel, then the show "Made in the USA" is a good survey of the varieties of manufacture in the US.
For amateur work:
Home Shop Machinist Magazine (this is for the forums)
Practical Machinist Forums
But there are probably hundreds if not hundred thousands of sites, texts, etc that you could read - you should see how many books I have on the subject...
And that would just be for metalworking, all these things grade into chemistry, woodworking, ceramics, computers, electronics, and all diverse arts.
In short it's sort of like asking what you would recommend for one starting out in philosophy...
I could literally write about this forever but I have to get to bed...
Although I never really do anything along these lines, I've always had a fascination with the sort of thing Nick does. I once had a dream in which I wrote a book (it says something about me that I dream mostly about having arguments and writing books) called How to Build a Civilization from Scratch in a Hundred Thousand Easy Steps, the ultimate how-to book, that dealt with gathering, hunting, building, cheesemaking, weaving, pottery, metalwork, woodwork, beekeeping, papermaking, all starting from nothing but natural materials. That would be an awesome book to research.
UPDATE: In comments, Nick recommends two more links: