Listen to Podcasts on the Apple Watch
Apple now includes the Podcasts app on the Apple Watch. This is a new feature on watchOS 5.
You can sync podcasts to the Apple Watch based on the ones you listen to on your iPhone when the watch is on the charger.
If you are mobile you can stream the latest episodes on LTE or on WiFi, but for the most part you will be playing from your watch, but thankfully you an stream when new stuff arrives.
Upgrade to watchOS 5 to get this new feature.