es lo mismo voy de compras seria go to shopping y me voy de compras seria i go to shopping queda mejor decir me voy de compras

I'd rather think of "me voy" as "I'm leaving (here) for"

Both are correct, if you want a difference then its this one.. Me "voy the compras" emphasizes the ME / I , you would only emphasize it if it's your own decision and you want to do it. If someone told you to / ordered to do it, you wouldnt say ME, you would just say "voy de compras". That is the case for this particular verb, its not like everything works this way tho :P

Is the same you only enphasize with ME at the beginning. VOY is trasnlated as I go, I'll go, I'm going