Appeal deadline questions after an unemployment denial

My claim was denied after 24 weeks of waiting, and the letter says I have a short window to appeal. I am in Hartford, in case local advice matters. 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 last year, and it settled down once I kept a simple log of every call and form.