It's a delightful game.
If you are self-employed, all you have to do is do the regular tax bracket income tax calculation on your net income (total income - work expenses) and add the fixed mandatory health insurance cost. It is very easy and can be done through e-Devlet or your bank's mobile app/website but many people just hire an accountant because mistakes can be costlier than the minuscule cost of hiring one. Another alternative is to get all your payments through a payment processor that handles taxes for you, which is the go-to option for side hustles.
If you are an employer, you do the same as above, but for yourself and all your employees.
For all the other extra taxes (car ownership etc.) you are notified if you owe anything and the payment is again as easy as online banking or e-Devlet. I've never heard anyone complain about the process of paying taxes. The absurd amount of taxes on the other hand...