Is there a way to keep the tasks more organized, based on categories?

Yes, assign a special colour to different categories, there are also power ups (some are paid ones though) to sort the cards.

There is a powerup Cardsync for Teams. That should group your cards for you.

I preffer to use labels as categories. They have colors and you can easily filter your tasks using labels. And the folders(lists) are always task statuses for me. For example folders(lists) can have next names, which represent task status: 1) To Do: task is created and ready to be assigned. 2) In Progress: task is assigned and in development. 3) Review: quality check of task results. 4) Done: task is complete 5) Backlog: sometimes I like to have a "backlog" list for the tasks, which are not yet planned but already known. I use this structure for all types of projects: conferences planning, software developmenr projects, master thesis writing and even my vacation planning. This helps me to have a clean overview of the project progress. It is also easier for others to see a snapshot of the project progresss.

You can add labels to tasks by hovering over a card and clicking the pencil icon in the corner and then “Edit Labels”. Once your cards are labeled you can filter them by clicking “Search Cards” in the menu on the right. This way you can choose to see only the cards in the categories you want across your whole board. Another option would be to treat your “Lists” as folders for different categories such as to-do, done, planning, etc. and move your tasks to whichever list makes sense for you.