Dont worry if your GMail inbox is full. You can clear it easily. Right now select all conversations option is being used. If you select and delete the conversations will move to the trash box.
But if you want to delete mails in selected sections, then select check box in the in box and click on the down arrow. You will notice read, unread, starred and unstarred items.
Select the section you want and then select the conversations to delete. The other issue is that the box will be filled with advertisements and promotions in the respective section.
Go to the respective box and then select one of the sections and go to the check box like earlier time. Then you can select the first 50 items. Then select all the 50 conversations. Then delete them.
All the mails deleted will be there in the trash section for 30 days. If you want to delete them you can do that in Trash folder also.