How can I get push notifications for emails in my inbox's sub-folders? By default it pushes just inbox emails, not the other folders that are in inbox.
You can't, the Gmail client for Android only syncs the Inbox folder. The other folders are only synced when you enter into each one.
If I'm not mistaken the K9 Mail client have the ability to sync more than one folder.
EDIT to add some new information provided by "Mr. Buster":
The current version of Gmail client for Android phones actualy have this option.
In order to configure the gmail client to sync more than one folder you have to go to:
Gmail Settings > *@gmail.com Account Settings > Sync inboxes and labels