Running is a stressful thing for the body.
The first week of C25K was hard work on my lungs - I thought they would burst!
I had a knee reconstruction a few years back after I tore my anterior cruciate ligament skiing. So for the first few weeks, my weak knee gave me a bit of curry.
My quad muscles needed a few weeks to get used to the whole concept of running and being used.
A year or so back I got bad plantar fasciitis from wearing thongs (flip flops, not underwear!) and standing all day teaching. Lots of ice, stretching and massage and orthotics fixed that. I felt twinges of it when I started running, but stretching it out has meant I've kept it at bay.
After I graduated from C25K and was trying to get a bit faster (I'm sooooo slow!) my calves were sore. I focussed on shortening my stride and really stretching them well after running and they seem OK now.
Now I am suffering from sore hips. I experience "sciatica" after housework regularly, but this feels like a combination of that plus my hip joint. Lifting my right foot from the accelerator to the brake is excruciating. Interestingly, it's not that sore when I'm actually running. I've Googled it (as you do), and I think it really sounds like piriformis syndrome. It's possible that is what my sciatica has been that all along.
I've found some good stretches and I'm hoping that this makes a difference.
And I'm hoping I don't continually find new and exciting ways to inflict pain on myself!!!