How can I do a close reading and analysis of "The Fish" by Elizabeth Bishop?

I think first of all you need to come up with some thoughts or interpretations of your own about the poem. For example, one interpretation is that the speaker seems to feel a great deal of sympathy for the fish. It has been caught at least five times before, and perhaps the speaker feels sorry for catching it again. Another thought about the poem is that there is lots of vivid, evocative description in the poem which creates a beautiful and tranquil scene.
The next step is then to reread the poem and identify any key quotations which will help you to support your own thoughts or interpretations. For example, if we begin with the interpretation that the speaker feels sorry for the fish, then we might cite the following lines:

I admired his sullen face . . . and then I saw / that from his lower lip . . . hung five old pieces of fish-line . . . still attached . . . grown firmly in his mouth.

In these lines, the phrase "and then" is important because it signals the change in the speaker's feelings about the fish, from admiring her catch to feeling sorry for it. The sympathy is suggested by words like "still" and "firmly." These words aren't necessary to the description but are there because they imply the speaker's emotional reaction. The fish-line "still" being attached, and lodged "firmly" in the fish's mouth, suggests how unnecessary the pain is, and how long the fish has been suffering for.
This kind of analysis of language is a key element to any close reading and analytical response to the poem. Try to identify key words and also figurative techniques (like similes, metaphors, and color imagery) in the quotations you choose, and then try to explain why you think the poet has used them. Or, alternatively, think about what effects these words and techniques might have on the reader. It's also a good idea to look for patterns of language in the poem, and to try to work out why the poet has used these patterns. Has the poet, for example, used similar colors to create a particular tone or mood? Or has she used any rhyme patterns, and if so, why?
At the end of your response you could offer an evaluation, summarizing your thoughts about the poem. For your evaluation, try to think about why the poet might have written this poem, and about how successful or otherwise you think her language choices have been.