For podcasts I recommend an app called Pocket Casts. Excellent UI, has all the featires you could possibly need including cloud syncing (and no bloat). No need to mess around with syncing to your PC etc. It is on Android and iOS and you can sync your 'cast subs across platforms.
For music - you could just copy your stuff on to your android device and then use PowerAmp (one of the best music players on there) to play it. But that's messy. I quite like Amazon's MP3 cloud locker service as an itunes replacement Google's is also good. Amazon's MP3 app allows you to download to the device and play the files with PowerAmp, Google's doesn't. I also found Amazon's PC software for uploading/matching your library to their cloud to be far superiour to Google's, which seems insane.
I have unlimited data though*, so I just got out of the habit of managing an 'offline' library, instead I just upload my stuff and stream it or download it if I will be without data for a while. It's a lot less elegant to upload to Google/Amazon on your wifi and then download again.
*Complete aside: I live in the UK, we seem to have much more competition in the mobile network market so plans are significantly cheaper here than they are in the US, including very affordable unlimited data. I have an HTC One which cost me about $40 up front, and I get (genuinely) unlimited data (including free LTE when it reaches my city) and loads of texts and minutes for $50 a month on Three. I just ran through their site and a similar deal in the US (with Sprint) costs in total around $250 up front and $110 a month, and as far as I can tell makes heavy use of throttling. So erm, basically you should all move over here.