What kind of fastener should I use to hang things on cabinets?
I would drill your holes for the rack through your cabinet, then drill some counter-sink holes on the inside of your cabinet so your bolts are flush with the inside.  You can find some flat headed bolts to use for this at any hardware store.  Have the head of your bolt on the inside of the cabinet and secure it with a lock washer and nut.  You should not need anything too thick as far as bolts go for this.  You idea on how to mount them will be fine.  The bad thing is that now you will have holes in your cabinet, so if you do not like it or if you want to remove it, you have just ruined your cabinets in a sense.