I use an app called
Downcast. I mostly listen to podcasts when I'm walking around doing things or out of the house, not sitting at my computer, so I don't do any kind of syncing. All my subscriptions are in Downcast and it dutifully grabs new ones for me semi-regularly. There are fairly robust settings for how long to keep episodes that can be adjusted for each podcast. For example, some podcasts I don't really listen to super often, but every once in a while I'll tune in and I don't mind if I miss a few. For these cases, I can tell Downcast to only keep around, say, three un-listened episodes and just purge older ones when new ones come in. I use this a fair amount to keep my queue under control (peace of mind!) and save space on my ipod.
Though, speaking of the queue, Downcast actually lets you have multiple queues, which it calls "playlists." It's a way of filtering your subscriptions. I have a Literature one with the various fiction podcasts I listen to, I have Talk Shows, Music, and I'm probably not really taking full advantage of the capabilities but either way it's still useful.
I have heard some people complain about the look-and-feel of Downcast being too clunky but truthfully I find it simple to use.
Pocket Casts is another app that I've heard about being good, but I haven't tried it. They have an Android version, which is nice if your phone is running that.