Got a denial letter for unemployment, what now? (update)

My claim was denied after 12 weeks of waiting, and the letter says I have a short window to appeal. There are 16 people counting on me, so I am trying to be careful. The letter lists a deadline and a hearing process, but it is hard to follow. If you have appealed before, what helped you prepare?

Thanks for writing this out. I went through something close to it 10 months ago, and it got easier once I kept a simple log of every call and form.