All the hoses and then the rad itself. Given that there's coolant in the battery tray the little hose from the rad to the expansion tank would be worth looking at first.
You can get UV dye to go in the coolant to help track leaks.
You'll loose some in the new filter. If your piston rings are all good then it might take a while to run down into the sump. Always fill, run for a few minutes and then top off.