What's the best tool to listen to podcasts offline?

This half answers your question. I think the problem of curated podcast lists isn't solved yet. At least I don't know of any. Discovery of podcasts for me still comes from a variety of sources. Mostly my own network for friends/acquaintances through PH, FB or Twitter. As far as listening goes. My favorite is Overcast. I subscribe to my favorites and have them auto download so that I can listen to them without worrying about having a good connection.

Breaker recently replaced Overcast as my default podcast player. While early, the app has all the table stake features you'd expect from a podcast app (subscribing, skipping, sleep timer, offline play, etc.) but with better, community-driven discovery.

I like this app for offline and streaming podcasts on my Android.