I love Player.fm
They have both an android app with a lot of good features (stream or download, can set how many episodes to download at a time, when to delete old ones, all per podcast and as global defaults, set up your own categories/lists, display order, play order, etc) and an actual web site where you can log in and listen to your stuff as well with synced history, so you can pick up pretty much at the exact spot you paused a podcast on your phone (or vice versa).
I am about to switch over to this setup from a Roku myself. I had seen recommendations for Flauncher, but I’m glad to see another launcher recommendation. I will have to try out Project Ivy. Thanks!