Three years later my iPod was replaced with an iPhone 3G but the syncing ritual was the same. Within a couple of years Podcast Apps started to come out for the iPhone. Some good, some not. I tried several over the years.
- The Podcasts App from Apple had a rough start but is better today. (And is now included by default in iOS 8.) It syncs across your devices with iCloud and is free so that's an easy way to get started.
- Instacast is one that I liked for years (until my current favorite came out) and aside from a few glitches a few years back has always worked well.
- My current Podcast listening App is Overcast. It's free, but spend the money on the in App purchase it offers to get the best out of it.