The first thing you might notice when you look at Billy Collins' "Introduction to Poetry" is the amount of comparison going on: the poem that the speaker is discussing transforms into many different things. This is probably a good indication that the dominant figure of speech is going to be either a metaphor or a simile. Both of these rhetorical devices draw a comparison between two things that are distinctly different.
A good rule of thumb to use when trying to figure out the difference between a metaphor and simile is that a simile uses "like" or "as," whereas a metaphor doesn't. However, it is this very distinction that is challenged in the first stanza of the poem:
"I ask them to take a poem and hold it up to the light like a color slide"
You might think that the presence of the word "like" indicates that this is a simile. But that isn't really the case. Don't look at the third line; just concentrate on the first two: here, the poem is implicitly being compared to a color slide -- without having to use the word "like." This is a good example of a metaphor hidden in a verb or predicate phrase such as "hold it up to the light." The speaker does not, for example, explicitly state, "The poem is like a color slide."
Stanza 2, although not a complete sentence, still contains a metaphor. However, this metaphor might be more difficult to interpret because the object that the poem is compared to is not even identified:
"or press an ear against its hive"
Collins is comparing the act of reading a poem to the act of listening for the sound of honeybees. Because the honeybees are not directly mentioned, this is called an implied metaphor. Since metaphors don't use "like" or "as" -- words that indicate a direct comparison -- metaphors are generally implicit by nature. However, when one of the things compared isn't even mentioned, you can make the distinction and call it implied.
As it turns out, every stanza of the poem contains a metaphor, but not all metaphors are created equal. There is something different about metaphors in the stanza 5 -- as well as stanzas 6 and 7: They are significantly more sustained than the metaphors in the first 4 stanzas. For example, stanza 5:
"I want them to waterski across the surface of a poem waving at the author's name on the shore."
Not only is the poem compared to a lake, but the fictional-readers -- the "them" -- are compared to water-skiers. The act of reading the poem is compared to the activity of skimming across the water's surface. The speaker would like the poem to be read, presumably with ease, grace and fun. As well, the author becomes a concrete entity -- a name written in the sand -- which the readers could have an affinity with. The speaker would like it if the readers would give a friendly wave to the author, rather than, you could say, ignore him.
The line between metaphor and conceit is not carved in stone: one of the distinguishing factors between the two devices is that a metaphor is relatively simple, whereas a conceit is "ingeniously elaborate." So, depending on your understanding of the word "ingenious," you could make a case either way.
This is also the case with the metaphor in stanza 1. You could, in fact, argue that it is a simile. So much of analyzing poetry depends on your ability to back up your argument with supporting evidence, and not on concrete rules.
The final evaluation? Since even a conceit is a particular type of metaphor, it is safe to say that metaphor is the overriding figure of speech used in this poem.