I won't get into all the details of my students' analysis, but I will say that they typically (as do all students, I am sure) struggle with Connotation (which is where the motherlode of ideas lies, I think).
With "A Bird Came Down the Walk," I showed the students how certain words--particularly the diction that deals with watery or nautical images, in the final stanzas, "oars, rowed, ocean, banks, plashless, swim"--tie together the ideas of the bird flying and the thoughts the speaker has of being in or on the water.
We discussed how humans long to fly, how swimming/sailing is peaceful and freeing, and yet--how flying and being in the water are both exhilarating and frightening. Both are joyous and yet also connected with dying and death, particularly rather violent ends.
This idea, suggested by the final lines of Dickinson's poem seems to indicate--to me, at least--that the speaker is fascinated by the simple life and beauty of the bird, and yet s/he doesn't, or shouldn't, long for its life, because it is one of danger and hardship. (Note the bird's "rapid eyes," described as "frightened beads." Wild animals always have to worry that they will be hunted, don't they?)
There is more to say, of course, but I'll leave it there. Poems are for discovery--but the point is that students shouldn't feel afraid, as wild animals do. They don't have to fear for their lives when it comes to analyzing a poem. It's not quite that difficult.