A few things for reassembly:
To apply the adhesive strips, peel one side of the backing off, then without touching the strips put the whole thing adhesive side down where the old strips were, then pull off the remaining backing. If you pull the strips out first, they want to tangle up and become a blob. Trust me, I know. Sadface.
To reapply the tiny bracket that holds the home button's cable in place, put the side with the one tab in first, oriented towards the bottom of the phone, then press the top with the two cut outs into place. There's two tabs in the phone body that should line up with the two cut outs.
When closing it up, you have to slide the top of the screen flush into place before pressing the rest of the phone shut. If you don't, the top won't snap shut.
If you end up having to pry the battery out, put some paper between the bezel and the battery. Mine now has a little gouge that's visible just under the volume buttons.