Blog index not going to wordpress folder

You should put the WordPress files in the root of the subdomain – that is, wherever your host’s document root for blog.mysite.com is. In my case, that’s /public_html/blog; but I assume that could be different from host to host.

I would recommend doing a fresh install, rather than just moving the files – unless you’re comfortable with potentially manually editing the DB to change WordPress URL/Site URL settings.