Google finally made podcasts official in Play Music last week, but not everyone is seeing the option even after updating to the newest app version. You might be able to give Play Music a kick in the butt by refreshing your account. It is an option in the app's settings—refreshing, not butt kicking.
This is a complicated process that will surely tax your technical abilities. First you have to open the Play Music settings, and then tap refresh "Refresh" under the account section. Pretty tough, right? Okay, but really, that's all you have to do. Go back to the main UI and check the slide-out nav menu. Podcasts will be there if the trick worked.
It's not working for everyone, but success has been reported by a lot of people. It even worked for Artem, and he has famously bad luck with technology.