I think that how long it takes to feel a difference in the way you describe depends very heavily on where you started and what your goals are. I played two sports in high school so my goals are pretty highly athletic - I'd like to get to the point that if you name an activity, I can physically do it at the level of at least a recreational athlete (obviously without possessing the requisite skills). Also, I always wanted to look, well, modelicious. So my other goal is to reduce my body fat, build muscle, or whatever is required to look darn good naked. Ten months after beginning intense exercise, I'm actually quite close to the first goal. I've made significant progress on the second, but it's going to take quite a bit longer.
I'd been exercising on and off my entire life so I doubt my time spans would really correlate. But it took about 3 months of consistent dieting and light exercise before I felt good enough to take on intense exercise, and by that time I had gone from like 240 lbs to 225 lbs and I could feel a significant difference - especially when I began to be able to fit into my older smaller waist size pants.
Since then, it was a gradual process of increasing fitness, but I never really felt any breakthrough since that three month time until recently, about one year since the start of the effort and ten months since starting to exercise intensely. The main reasons I feel as though I've made a breakthrough are that:
-I've grown my commitment and my fitness enough that I can now work out intensely six days a week and not feel completely wrecked.
-I recently added serious weightlifting to my regime. Not only could I not be more pleased with the results even in two weeks, but experience tells me that will definitely eventually do exactly what I want. On top of all that, I love how it makes me feel even within a few days of doing a single workout. Real life feels so EASY compared to pushing around heavy blocks of iron.
-Of course with all that working out, not only do I have to eat more, but my metabolism is faster. I am warmer all the time, and I can tell. It feels like there is just more energy coursing through your body when you get there, and it is way cool. So just the feeling is different, and that I think is the real breakthrough you are talking about.
FYI, I don't think I can even run 5k straight at what I'd consider a respectable pace.. maybe only very very slowly. I wouldn't consider running ability to be the only measure of fitness or health that matters.
I think that you will feel a truly significant breakthrough, physically, when you lower body fat percentage to a certain point. My guess as to when this would be for most would be about 25% - which, interestingly, dovetails nicely with the top of the American Council on Exercise's "average" range. en.wikipedia.org/wiki/Body_fat_percentage#
If you do some toe point-and-flex regularly, pulling and pushing with as much force as you can muster, I believe you can exercise those shin splints away. Or at the very least reduce them or make them go away faster.