How to only include certain Podcasts on the iPhone with the Podcasts app
EDIT: With more recent iOS updates, this technique is no longer as useful. Playlists are useful to control whether an episode is sync'ed to the device. If a podcast is in a playlist, and that playlist is selected to sync to the iOS device, then all the podcasts in that playlist will be transferred to the device. 

What no longer works is being able to create a playlist in iTunes and then play that playlist on an iOS device. There is only a single Up Next playlist available, and you have to manage it on each device.

ORIGINAL: I use Podcasts on my MacBook Pro, my iPhone 5 and my iPad Air 2. The only thing that I have found to work is this setup.

All my downloads of new episodes, etc. are done on the MacBook Pro. I basically leave iTunes running most of the time even when I am away and only have my portable devices with me.

I use Playlists on the MacBook pro iTunes to sort episodes for which device. Video podcasts primarily go to the iPad and audio primarily to the phone, but when flying I often put audio podcasts on the iPad to save battery life on the phone.

Then in iTunes I connect each device and use the settings for the device to control which podcasts and which playlists are synced to the device. I generally only sync checked files and I uncheck episodes after they are played.

Also in iTunes on the MacBook Pro I have smart playlists for recently played and recently added and I use those to help me find the newer Podcast episodes quickly and to uncheck played episodes quickly.

This is admittedly a lot of work. I spend a couple hours or more each week managing my playlists. I sync nearly every day to add new episodes and clean up old episodes. I just could never make the "automatic" sync'ing work the way I wanted, so I decided to take control of it with these techniques.

Final Note: in iTunes Preferences on the Store tab, I turn off Sync Podcast subscriptions and settings. I have found there is a bug in this process that sometimes causes iTunes to stop downloading any podcasts. There was some discussion on the Apple Support Forums that pointed me to this change.