The thinner black hoses go to the heater core. Remove the hoses from thermo housing and other, and then flush water in one side, and then switch and flush into the other side (have the hose on pretty high, yes you'll get sprayed).
You should be good flushing both ways. I don't recommend putting anything in there, CLR etc. just use water.
If you are getting a radiator from mopar, ask them, if you not sure what coolant to use. I got my radiator from the dealer too, and was able to haggle on the price. It is much better one than after an market, IMO.