A milk run is an easy or uneventful task, especially a task requiring travel. It is often used in aviation, to refer to an easy flight; either one that is short or so routine that it poses no challenge. This meaning was firmly rooted in the public mind during WWII, when fighter pilots referred to easy missions as milk runs.
The term almost certainly comes from the daily deliveries of the olden-day milkmen, who traveled a routine milk route every week. Mirroring this, many people use 'milk run' to refer to a trip or route that involves many stops -- a shopping trip where one stops at many stores, or a train route that has many stops (in contrast to the express route). While this usage is currently falling out of favor, it is possible that this usage predates the 'easy task' meaning.
In decades past this was also sometimes used to mean the early train, as that would be the train collecting milk from outlying farmers and bringing it to the towns. This meaning may still be used in parts of the UK, but I have not come across it in the US.