This is how I set up syncing podcasts to my phone. It gives me a single playlist that I can just listen to straight through with everything in order, and it won’t remove them until I’ve actually finished them OR gone into iTunes and right-clicked, then selected “Mark as Not New.” If I’m in the middle of a podcast when I get home and sync the phone, the podcast won’t be removed from the device and I can resume from where I was either in iTunes or back on the iPhone. (This should work on all iPods, not just the iPhone.)
Apologies if this is obvious, but it took me some time to figure out the right settings. I was tired of only making it halfway through a podcast and then having it disappear out from under me just because I attached my phone to my computer when I got home.