I've read a bit about tarry, shiny creosote, and I don't have that. What I have is a thin coating of soot and some flaky pieces here and there. Is this something I should worry about, or just start burning in my fireplace again this season like normal?
Ash is not flammable, and ash can sometimes be in flake form. If that is what you have, there is no danger.
Even if there is some creosote buildup, that is not necessarily dangerous either. What is dangerous if there is enough heat and creosote thickness for the creosote to burn long enough that it overheats the chimney causing structural weakening, or emits sparks which ignite combustibles outdoors like a wood shake roof, trees, shrubbery, or other structures.
If creosote buildup is minimal (less than 1/4 inch), it can be safely removed by building a hot fire to burn it out. If you aren't sure about doing that, don't hesitate to bring in an expert: either a firefighter or a chimney professional, though the latter tend to have a conflict-of-interest.