First, find a podcast using the "Search" tab. Then tap "Follow". The podcast will appear on the Home screen in the "Following" section.
Any time there is a new episode, you'll see a blue dot on the podcast cover.
When you tap inside the podcast with a blue dot, you'll see which episodes are new since you viewed the podcast last time. They'll also have blue dots.
See also How frequently does Metacast check podcasts for new episodes?