how would I add their data in the tables I made in the database?
By using JDBC to send INSERT statements to the database, one for each object in your object graph (assuming you have one table per class).
And yes, that's a lot of boring, repetitive code to write, which is likely to contain errors and quite a burden to maintain.
Which is why people have written Object-Relational mappers like Hibernate, and there's a Java standard for that called JPA. Which is part of Java EE, but that doesn't mean you have to run a Java EE server to use it.
Update: OK, so you cannot use an ORM, probably because you're supposed to understand how JDBC works first, which makes sense because ORMs are based on JDBC and sometimes you have to go down to that level when there is a problem.
When using JDBC directly, you typically create a DAO (data ccess object) layer that is responsible for writing and reading objects to/from the database. For dealing with a nested object, one DAO can call the other.
And no, the DAOs are not called in the setters and getters. There called by the part of your application that reacts to user input. E.g. when the user opens the application and the start screen shows a list of employees, you'd call an EmployeeDAO.findAll() method, and when the user makes changes to an employee and clicks on "save", you'd call the EmployeeDAO.save() method.