Why is a topology made up of 'open' sets? Part II Because the display was getting quite cluttered, I thought I'd post a second part to this question separately. I hope the Gods of Math Overflow don't take too much offense. I'll  go now into some details with which I wished intially to avoid prejudicing the replies.
Here are three natural areas of progression in the study of topology:
(1) point-set topology;
(3) sheaves and their cohomology;
and a very important 
(2) middle ground 
that I won't give a name to, involving matters like the classification of two-manifolds. I don't quite agree that the metric space intuition is sufficient even for a first course, simply because we always look ahead to the way the material will help or hinder students' future understanding. 
Now, it is in (3) that a topology in the abstract sense plays a very active role, as the relation to global invariants emerges prominently. At this point, the definition needs no other motivation. However, in an introductory course on (1), I have yet to incorporate successfully interesting material from (3). When dealing with (2), curiously enough, the definition of a topology plays a very passive role. Many arguments are strongly intuitive. However, it is clear that one needs to have already absorbed (1) for the material in (2) to feel really comfortable. Of course there are exceptionallly intuitive people who can work fluently on (2) without a firm foundation in point-set topology. But for most ordinary folks (like me), the rigor of (1) is needed if only as a psychological crutch. Note here that  most of the natural spaces that come up in (2) are in fact metrizable. However, getting bogged down with worrying about  the metric would be a definite hindrance in working through the operations that come up constantly: stretching, bending, and perhaps most conspicuously, gluing. A quotient metric is a rather tricky notion, while a quotient topology is obvious.
With this future work in mind, how best then should one get this background in (1)? When I was an undergraduate, in fact, there was plenty of motivation in  course (1). We used the first part of Munkres' book, and had exercises dug out of 'Counterexamples in Topology,' involving many strange spaces that have one property but not another. It was great fun. You may wonder then, what exactly I'm worried about. It's that I felt later that this kind of motivation had not been quite right. It wasn't entirely wrong either. Certainly we became very confident in working with the definitions, and that was good. However, and this is a big 'however,' when I moved on to (2), I had a distinct sense that the motivational material and exercises used in (1) were actually preventing me from learning the new notions efficiently. It took me quite a long time to make the transition,  bugged by a persistent longing for the axioms and the exotic examples. A number of conversations I've had over the years indicate that my experience was far from unique. At any rate, this experience makes the issue for me quite different from a course on, say, linear algebra. No doubt the motivational material for diagonalization used in most courses is hardly convincing. Substantive examples come later, sometimes much later. But most of the operators and eigenvectors in standard textbooks are good toy models for a wide range of serious objects.
One obvious remedy would be to incorporate material from (2) into (1). I found this very hard, mostly because of the point already made above, that the role of (1) in (2) is quite implicit. When I first posed the question, I was hoping someone had figured out a good way of doing something like this. Incidentally, sheaves came around much later than the definition of a topology, so the historical question remains as well. How were the standard properties decided upon?
Let me make clear that I am not arguing that everyone has to go through the progression just outlined. Obviously, some people will take (1) elsewhere, in a way that the issues become quite different. Perhaps many people will never need more than metric spaces (or normed spaces, for that matter). But (1)-->(2)-->(3) is certainly common enough (perhaps especially for arithmetic geometers) to call for some reasonable methodology.
Meanwhile, I also appreciate Andrew Stacey's point (possibly even more than he does!). The long paragraphs above notwithstanding, the pedagogical question isn't something I lose sleep over. But it would be nice to have a few concrete and systematic ideas to use. They would certainly help me to understand the subject better!
Added:
Perhaps I should rephrase the question: How should we teach point set topology so as to facilitate  the transition to the topology of natural spaces?
Somehow, this way of putting it seems much vaguer to me. 'How to teach X?' is such a broad question I would never be able to answer it myself in a finite amount of time. It seems to invite a diffuse discussion of everythng under the sun. That's why I had preferred to focus on one rather precise mathematical facet of that question. But I don't feel too strongly about it either way.
 A: I wonder if your middle ground is "that which we study" versus "that which a topological space is." So many of us who call ourselves topologists study: (1) surfaces, (2) knots, (3) curves on surfaces, (4) surfaces that bound knots, (5) embeddings of one manifold inside another, (6) defining quantities on all of the above that are invariant. 
Invariant with respect to what?!? Well, with respect to the underlying topological structure, of course! 
How can we study these? Often we draw pictures to indicate that even if we had a formula for the curve on the surface or for the embedding of the knot in space, the formula would not help gain a global understanding. In an sense, topology as a study is the study of things for which we would rather not ascribe quantitative details when qualitative details are adequate. Before, you jump on me: winding number, fundamental group, 3-dimensional volume, etc, etc, are quantitative details. BUT they are quantities that do not depend on the quantitative description of our space or our embeddings. Rather, they are independent of the particular description. 
It has been rare in my career, outside the classroom, that I have actually had to demonstrate that a map is continuous. When I draw a knot diagram, I believe I have the wherewithal to show that there is a function from a circle into space that has the properties that the picture indicates. Similarly, when I draw a pair of loops encircling the knot, I believe that I have the skill to parametrize them, and to parametrize their path composition.
So even though the notion of open sets, arbitrary unions, and finite intersections rarely come into play in the work that I do, they provide the backbone of the subject  --- that which allows us to work at the intuitive level. 
I haven't read the details of the first thread, but I imagine that someone explicitly mentioned the quotient topology to develop the notions of sewing or gluing spaces together. In the middle ground, or in the world in which we work, we can get away with the techniques that were called "hand waving" in the last century because the mathematicians of the last century defined topology in precisely the right way --- the way for which our intuition matches the rigorous notions.
A: Your question reminds me of Grothendieck's Esquisse d'un Programme, specifically the notion of "tame topology". Was this on your mind when posing the question? I am thinking of the part that starts like this:
After some ten years, I would now say, with hindsight, that “general topology” was developed (during the thirties and forties) by analysts and in order to meet the needs of analysis, not for topology per se, i.e. the study of the topological properties of the various geometrical shapes. That the foundations of topology are inadequate is manifest from the very beginning, in the form of “false problems” (at least from the point of view of the topological intuition of shapes) such as the “invariance of domains”, even if the solution to this problem by Brouwer led him to introduce new geometrical ideas. Even now, just as in the heroic times when one anxiously witnessed for the ﬁrst time curves cheerfully ﬁlling squares and cubes, when one tries to do topological geometry in the technical context of topological spaces, one is confronted at each step with spurious diﬃculties related to wild phenomena. 
To me the notion of a "false problem" is a very succinct way of expressing the problems associated with going from (1) to (2) in your formulation, at least if I have understood your question correctly. Namely, in a first course in point-set topology, one spends a lot of time learning about topologist's sine curves, long lines and Sorgenfrey planes, and the problem is that all of these are cooked up to solve/exemplify problems that are "false", i.e. not very natural/interesting to someone who actually does geometry in his day-to-day life. 
If we then believe Grothendieck, the problem is not one of didactics at all: the difficulties in passing from (1) to (2) are an inevitable byproduct of the fact that point-set topology is not adequate as a framework for geometry. Maybe this is true. In a first topology course, one does need to spend time on pathological behavior, simply because anyone who does topology should know that a topological space with no extra conditions can be weird and not very geometric. I think this will be true as long as topological spaces are considered the basic framework for topology. Then one does have to unlearn the instinct to think about pathologies when actually doing topology. On the other hand, of course you can't tell students in a first topology course that the pathological spaces they are learning about will not actually be so useful later in life; no one would learn the material after hearing something like that. I suspect that I have just restated what you said in a clumsier manner. 
A: If your main concern is just weaning students off of metric intuitions, then it seems to me that introducing students to topologies on partial orders like the Scott topology will do the trick. You can use it to give them a running example which is (a) practically useful, since it underpins an enormous amount of research into formal semantics and computer science, and (b) full of features unnatural from a metric/geometric point of view. (Furthermore, if you do want a more geometric example of the relevance of the Scott topology, you can use it to explain semicontinuity.) 
A: Even if your area of study is normed spaces, it is not enough to know just about the topology derived from the norm. Weak and weak* topologies play an important role too. And different topologies are important in PDEs. 
If I am teaching the concept of a topology and trying to defend it against an imaginary critic who says "All interesting topologies are metrizable" then I'll emphasize the notion of a product topology. Actually, one of the nicest examples, the product of countably many copies of {0,1}, is metrizable, but it's easy to see (i) that there is no single metric that is obviously best and (ii) that it is somehow nicer to argue directly from the topology.
PS I found algebraic topology very hard when I first met it. I think it was for exactly the kinds of reasons you describe. 
A: Here are my reactions to the latest:
1) I think for the vast majority of people who work in differential topology and geometry and global analysis, the idea that topology expresses the notion of "closeness" without using a specific choice of a distance function is enough. In fact, for most of us, everything is locally Euclidean. This is certainly good enough for classifying 2-dimensional surfaces, which leads to my second thought...
ADDED: Response to Minhyong's comment: I don't mean that any metric should actually be used. The whole point is that you want to study the properties of a space (here, a smooth manifold) without imposing a specific metric on the space. To me, the idea and definitions of topology arise naturally when you want to identify the properties of the space that remain valid, no matter which metric you use (changing the metric corresponds to stretching or warping the space).
2) I don't know what you mean by the middle ground. Could you elaborate? My ignorance is almost certainly due to 1).
3) It seems to me that, Cech cohomology notwithstanding, the progression to sheaves is critical only if the topological space has nontrivial local structure to it. My impression is that sheaves play a central role in subjects such as complex analytic, algebraic, and arithmetic geometry, because singularities are unavoidable yet tractable. They are seen much less in the smooth real category, where singularities are usually avoided and, despite the efforts of Thom, very difficult to deal with.
ADDED:
4) I see now that Minhyong's question is really about teaching a point-set topology course. I am ill-equipped to answer properly, because I have never taught such a course. I did take one at Penn, while I was an undergraduate, and I have to say that, although the course was taught very well, I found it even back then to be rather pointless. Nothing has changed my mind about this since then. For me, I find topology to be a compelling subject only when it is modified by adjectives such as "differential", "algebraic", "symplectic", "arithmetic" (?), but not when it is unmodified or modified by "point-set".
