Does the speaker's mood change as the poem continues? If so, where?

The first six stanzas are a celebration of the beauty of the skylark's song and the freedom of its flight. The skylark sings only when flying, and the bird is flying so high that the speaker can no longer see it. Given that the skylark is invisible (but still heard) and so close to "heaven," there is an immaterial, spiritual quality to the skylark's song. So, it is not just the beauty of the song that affects the speaker; he is also struck by the spiritual symbolism of the experience. Note that in the first line, though, he calls the skylark "blithe." This means happy but indifferent to (or unaware of) the suffering of others. These first six stanzas are a celebration, but there is that initial tone of envy as well. The speaker envies the skylark's ability to feel so happy and free, whereas he is burdened with his own problems.

In the next six stanzas, the speaker shifts from this celebratory-yet-envious tone to a more pessimistic tone. The speaker cannot find the same earthly beauty as he had experienced with the skyward skylark. Thus, the tone shifts as the speaker switches from describing spiritual beauty to earthly limitations. In this stanza, he notes that the rainbow's beauty does not match the skylark's song:

 What is most like thee?

         From rainbow clouds there flow not

                Drops so bright to see

As from thy presence showers a rain of melody. 

In these six stanzas, we have other examples of things unseen or unnoticed. Whereas the speaker did notice the unseen skylark's song, these examples go unnoticed. There is the poet's "unbidden" (not asked for) hymns, the "lovelorn" maiden in the tower, the glowworm's hidden hue, and the rose clouded by its own leaves. The tone is downright melancholy at this point. Here are all these wonderful things which go unseen, unheard, and so on. 

For the remainder of the poem, the speaker asks the skylark how to be a natural poet. In the skylark, he sees unlimited potential. He, on the other hand, feels limited by his earthly concerns. This symbolism is illustrated in the first two sections of the poem. The first 30 lines express the skylark's beauty and freedom. The next 30 lines express the poet's notions of earthly limitations. The remainder of the poem shows how the speaker would like to be a poet with the same potential and beauty that he perceives in the skylark. So, the tone shifts from celebration and envy to pessimism to plaintive (pleading) requests.