+1 on Chi Running and not too much shoe. If you don't get started with the built-up shoes then you don't have to wean yourself off them.

I think there are a lot of us here who started running in our late 40s or early 50s. I can't speak to 60 yet, but why not try? There's plenty available about injury prevention (a lot of it boils down to strengthening your feet and your hips), and I don't see why that should be any different at any age - younger people heal more quickly and can sometimes get away with more sloppiness, but it's really not an excuse for them to continually injure themselves.