Hi - I was at the Piazza SS Annunziata this year for the New Year procession. Surprisingly, the tourist office was not helpful with details so we just showed up - I will try to check the time we went and post again bit I think we went around 10:30 or 11 and had to wait for a bit.
There is a small market, mostly food and candy, in the piazza - not really anything special. The procession is fun to look at - people are dressed as you suggest, in costume and there is drumming. It's nice for photos if you haven't seen a procession like this in Florence before (they have a lot of them I think).
The parade people end up in the church. The street in front of the church is your best view I think.