Whether to use "get" or "collect" is a matter of depth of meaning. Although both are interchangeable in terms of retrieval, to "collect" may also convey to gather and perhaps hold onto for a time.
Get: to go somewhere and bring back someone or something
Collect: 1. [mainly UK] to go to a place and bring someone or something away from it
- to bring something together from different places or over a period of time
Since the example asks that something be picked up and kept for a week while the person asking for this favor is away, "collect" might be the deeper meaning of the request. However, someone knowing that the person asking will be away will likely assume that holding onto the package is involved. In this interpretation, "get" also implies gathering and holding.
As to your question of common usage, in the U.S.--"get" and in the U.K.--"collect".