"What type of research design should a marketing researcher use to find out how many customers there are, what brands they buy and in what quantities, which advertisements they recall, what are their attitudes toward the company, and who is the competition?'

Descriptive Research.

Explanation:

Descriptive marketing research aims to observe and discover some market phenomena, so that it is possible to describe, classify and interpret such phenomena, such as what is happening in the market, specifying who, what, where and how.

It is usually carried out through discussions, questionnaires, interviews and specific groups.  It serves as a basis for explaining the occurrence of such phenomena encountered.