Podcast syncing between iTunes and iPhone doesn't work
Apple's entire podcast ecosystem is known to be buggy and has been for a long time. (See, for example, this bug of utter disconnect between iTunes and iOS prior to the arrival of the Podcasts app. It seems the point of moving podcasts to a dedicated iOS app was simply to banish the irredeemably buggy podcast code from the otherwise well functioning Music app code. I still have the exact same problems with iOS 7.1 and the Podcasts app.) You're not doing anything wrong, the stuff just doesn't work.

You will have better luck with Instacast. There are other third-party podcast players for iOS, but AFAIK Instacast is the only popular one that also plays on Macs and syncs between iOS and Macs. As of this writing (April 2014), Pocket Casts is arguably better than Instacast with respect to syncing among mobile devices, as it even syncs between iOS and Android, but it does not have a Mac application to allow you to play podcasts on your Mac.