I've been trying to start running, but after a couple 20 minute sessions I started getting really bad pain in my calves and occasional muscle spasms. I can still walk for a few miles at a time without it getting too unbearable, but it's quite painful for me to run now. This is not normal soreness, but I have no idea what it is.

Does anyone know what this could mean? I seem to be prone to minor running injuries. When I was thinner I ran more and ended up with (what I thought was) a stress fracture, but I had to take painkillers constantly for assorted knee and ankle pains.

Maybe it means don't run - it's pretty high impact on your knees!

Try walking, swimming, elipticals, etc.

And also try taking Folic Acid (get at Whole Foods for like $3). Helps noticably with muscle pains!!

If you're only starting again (regardless if you've ran before) then I'd say you're going too fast, too soon.
Try the couch to 5k program maybe?

Not sure the cause but agree that perhaps this is a message from your body to either not run or start very slowly at running. There are so many other fantastic types of exercise out there, I would try something different for a while.

I kind of had the opposite happen to me. I hurt a little after walking... Just could be your body adjusting.