You can use both Story and Task boards. Task board will give you a bit more detailed view on the progress of your stories. You can setup the status workflow for both stories and tasks as you wish in the project settings. Also you can specify Work in progress limits for each state on both types of boards.
But keep in mind that in Agile and in Kanban you want to deliver new features (aka stories) to your customers as soon as possible. But not not half backed stories that only have 50%of their tasks done. Only truly done work. So tracking the story progress on task level could be a bit misleading as it wont give you enough context - a single story could be on track but many more could be waiting to be implemented or stuck somewhere else in your process.
So you can setup a board with all the stories, bugs, feedback etc you are working on and monitor the progress on the entire workload of your team. The Work In Progress limits on each column of the board will allow you to see if there are problems and bottlenecks at certain stages of your development process so you can make adjustments
We have a section in our documentation regarding Kanban:
Check it out and see if that works for you.
As for the automatically setting the state of the story when all tasks are done - we have received similar feedback but still haven't implemented it. Check out our feedback portal, vote and share what would work best for you there: