You can listen to podcast directly on the website here or you can subscribe to it through one of the below avenues. This will let you know when a new podcast is published as long as you turn on notifications within the application.
iOS ships with a default Podcasts App. To subscribe to Dragonscast through iTunes, simply search within the App for Dragonscast. Once you see the result for the App under Podcast Episodes, click it and then on the new screen you should see an option select Subscribe.
Android does not ship with a default Podcasts App but there are plenty of third party options available. Additionally, you can also use the Google Music App to search for Dragonscast and choose to subscribe through it if you wish.