Android: There's no shortage of great podcast managers for Android, but if you're looking for a solid app that won't cost you a penny, AntennaPod is worth a look. The app packs support for OPML imports and exports, subscriptions via RSS and a homescreen widget player.
The app covers all of the basics: you can bring in your podcast subscriptions from any app that lets you export them as OPML, or you can subscribe manually via RSS or ATOM. The app's search tool isn't that great, so you're better off getting the URLs for your favourite podcasts and using those instead. Once you are subscribed though, you'll find AntennaPod supports audio and video podcasts, has a sleep timer so you can listen at night, supports both downloading and streaming of podcasts and sports a great-looking Holo UI with day and night themes.
Admittedly, you're not going to find some of the more advanced features here (variable-speed playback, subscriptions via password protected feeds, etc) that you'll get in our favourite podcast manager, Doggcatcher, but AntennaPod also doesn't come with its $US5 price tag. If you want simple, free podcast management, AntennaPod is worth a download.