The built in birthday calendar takes it's data from the Contacts app, and you have to add the birthdays there. If you want a Birthday calendar that you can edit in the Calendar app you'll have to create one from scratch.
If you are using GMail, as your post suggests, you'll need to set your GMail account up as an exchange account, and follow the Multiple Calendars setup. Instructions for both are in the below link.
Setting up Google Sync with your iOS device - Google Mobile Help
You won't be able to create new calendars for GMail on the iPad. You'll have to do it on the Google Calendar site, and then use the syncing instructions in the above link to sync it to the iPad. You can add/edit/delete events as normal once it is set up.
If you are using an iCloud account, or just local calendars tap the Calendar button, then the Edit button and choose Add Calendar
I don't recommend using the Calendar app locally only. It is easy to lose important data when it is stored in just one place, especially a mobile device.