After your tree has been felled, there should either be a log or a felled tree in the tile where the tree once stood. If there is only a log, right-click it and take it. Otherwise, right click the felled tree and select Chop Up, which will produce a log each time until it disappears. You should only require one log, so take one.
Here is where you can either take the skill-improving route (recommended), or the quick route. For both routes, you will need to active activate your carving knife.
If you chose the skill-improving route; right click a log and hover over the Create sub-menu, then choose a random option and keep repeating that after each action is completed until you have a pile of wood scrap in your inventory. If you have more than one scrap, there are 2 options: right-click the entire pile and choose the Combine option (which can combine up to 10 or 11 scraps), or activate one scrap, right-click another scrap, and choose the Combine option.
If your wood scrap has a weight value of 1.5kg (the standard measuring unit, if you don't have this much then make more scraps and combine them), activate your carving knife, right-click the scrap, hover over the create menu and choose kindling. Keep repeating this process until you have a kindling.
Activate your carving knife and right-click the pile of wood scraps that you just combined. If it is bigger heavier than 1.5kgs then make kindling out of it. Repeat this process until you have at least 1 one kindling.
If you chose the quick route; right-click a log and hover over the Create sub-menu, then hover over Miscellaneous and select kindling. If you make kindling, continue to the next step. Otherwise, keep trying.