I started running mixed with walking a year ago.
Because I had never exercised for more than 10 years before that, I started from a very simple step. The very first time I did last year was I walked for only 60 meters and no running at all, and that's it, I got home and tired already. But that's my history, now I can walk more running actually than walk for 4 km and I often still want to do more and more when I got home. I can also walk in my weekend morning for 15 km and I don't feel hurt on my feet.
I continued exercising persistently, being disciplined to exercise 2 - 5 times a week.
I feel good, fresh, fit and more productive. My body feels lightweight, being from 110kg of bodyweight last year to 75kg now. But I don't think I grew my muscles that much on my feet.
I tried You Are Your Own Gym (YAYOG) by Mark Lauren 3 weeks ago, this book contains 111 clearly illustrated exercises, including muscles for feet. After practicing it for 3 weeks, I know my glutes (buttocks and hip flexors), quadriceps (front of thigh), hamstrings (rear of thigh), calves muscles being stretched and stronger. I couldn't do squats the first time I tried it I felt dizzy, I think that's because of my Vertigo. But I kept trying verrry slowly to go down as far as I could, I only could go down for about 10 cm as far as I remember, if I went down lower than that my kind of vertigo dizziness attacked me. But I'm happy to say that is also my history now :-) Now I can do squats exactly like Mark's photo example from his book. I know I'm sure my body movement is correct because I can feel the right muscle being stretched. Just to mention one of many creative exercises to strengthen feet muscles from YAYOG practice.
I combine running almost every morning and YAYOG to strengthen my body. As I'm still early on YAYOG, so I'm not sure, but I feel that my feet muscles help me lift my body to run/walk more easily than ever when I run/walk. My balance seems to be improved.
P.S. Oh, I actually want to add additional comment for Berin's answer above me on barefoot running, but it seems I can't do that because I don't see the comment link as I'm used to see on stackoverflow site, just make sure the road is good and safe enough to do that, I did barefoot running several times a long time ago, but then I stopped because my doctor adviced me to keep wearing my shoes on. She said some of her patients got diabetes not long after they stepped on nail on the street.