The instructions are a guide only, Where it says tumble for a week, ignore, tumble for as long as needed. First stage is a coarse grind that will remove sharp edges and smooth the rocks, it's usually something like 60 or 80 grit. Agates Petrified wood will need to be in the first stage for many weeks even months.
Every stage after that is removing marks left by the previous grit, not removing lots of material, second stage 220 grit Silicon carbide for about a week. Third stage 600 grit Silicon Carbide, keep the rocks in this stage until they have a dull sheen about them, it could be several weeks, the grit breaks down from hammering from the rocks to finer and finer until it is giving a pre polish.
Polish stage, they say Aluminium Oxide, if you can get it it needs to be 14000 grit. Tin oxide is what I use, it's expensive but it doesn't require much to work, I'm not sure what size barrels that tumbler has but try a teaspoon of Tin oxide and maybe half a cup of water for about a week, run a bit longer if you can.