Browse to any podcast and then tap the Episodes tab, from there you'll see all the episodes for the show. Some shows like those with a news focus will display the most recent episode at the top, while others with a serialized format will display the first episode at the top.
Following a show means new episodes from that show will appear on your Home screen. You can also set things up so that new episodes will be automatically downloaded, and even have them added to your queue.
Episodes from The Daily on iOS
New from shows you follow on Home for iOS
Viewing new episodes from shows you follow on iOS