In this context, "pick up" means buy or get. It usually, but not always, is an incidental purchase while traveling from point A to point B ("on my way home," "while I'm out").
I will pick up some food on my way home.
I will get some food on my way home.
I will buy some food on my way home.
Tomorrow, I will pick up a cake for the party. (= Tomorrow, I will buy a cake for the party.)
It generally applies to small items that are purchased routinely without thought.
I will pick up some paper for the printer.
I will pick up some diapers for the baby.
I will pick up a newspaper.
I will pick up a monthly bus pass.
It does not apply to purchases that require significant thought.
bad - I will pick up an airplane ticket to Paris. (thought is required - which day? which class?)
good - I will pick up a monthly bus pass.