How do you go about... asks how someone starts or handles a task. The task is named with an -ing word after the verb: go about finding, go about doing.
How do you go about getting a library card? asks for the method, not yes or no.
When the verb that follows takes the -ing form, go about find is wrong; say go about finding, as in How do you go about finding a job in a new city?
It asks how someone starts or handles a task, naming the task with an -ing word: go about finding, go about doing. The question wants a method or the steps someone actually took, not a yes or no answer.
Planning a move to a new city, you might ask a friend who has done it: How do you go about finding an apartment there? You are not asking whether it is possible, you want their method.
At work, a new task arrives and nobody knows where to start: How should we go about organizing the files? The -ing word organizing names the task, and the question asks for a way of tackling it.
A nearby distinction: the verb that follows takes the -ing form, so How do you go about find an apartment? is wrong. Say go about finding. A close lookalike, go over, means to review or examine something, which is a different meaning entirely.
How do you go about getting a library card here?
That question asks for the method, the actual steps, so a useful answer would describe them rather than just say yes.