How do you detect if a part's position has been changed?

Use Changed. This even fires whenever property of an instance is changed, like position!

script.Parent.Changed:connect(function(pos)--You need a function
	if pos == "Position" then --if pos is position then
		print("PartMoved")
	end
end)

If you want to record the old position and the new one try this script:

local oldpos = script.Parent.Position
local newpos

script.Parent.Changed:connect(function(pos)
	if pos == "Position" then --if pos is position then
		if newpos ~= nil then
			oldpos = newpos
		end
		newpos = script.Parent.Position
	end
end)

Hope it helps!