I'd strongly recommend Pocket Casts. One extremely nice feature is that if you have multiple devices (including iOS ones) it will remember where you're up to in each show across devices.
The other killer is that they manage the episode refresh check through their servers so it's a single operation for your device to check for new stuff, instead of it having to go out and check (in my case >50) individual sites like other apps.
Sadly, you cannot copy the MP3s off an iDevice using Pocket Casts. I wanted to build an archive of a couple podcasts, and lost months of episodes when I wanted to replace my iPad.
This is probably to late for your use, but there is a neat Mac utility called iExplorer that lets you browse your iDevice's filesystem and copy files off of it. I'm pretty sure you could have grabbed the files using a trial of iExplorer.
The other killer is that they manage the episode refresh check through their servers so it's a single operation for your device to check for new stuff, instead of it having to go out and check (in my case >50) individual sites like other apps.