Use calendar.module in my custom module
I would second the comment on trying out Views. You should be able to achieve what you want with the menu tabs and a specific calendar on that page with Views. Maybe try checking out the documentation. Without more detail in your question it is hard to say, but I'm pretty sure you don't need to make a custom module to do what you have written so far.

Also you probably should also specify which version of Drupal you are using. I'm assuming 7 but it is always better to let people know.