I would like to add a new service which will use another class (say, class A) to complete the task. What is the procedure to add a new service in aspnet zero
You can add a service just like you do in a regular ASP.NET Core app. I think, by service, you mean a new project. If so, you need to create a module class in your new project (see https://aspnetboilerplate.com/Pages/Documents/Module-System). Your new module must depend on your other module which contains class A.
Please let me know if you face any problems while doing that. You can also take a look at Migrator project in the solution as an example.