As far as I know there is no automatic downloading on the iPad. iTunes on the computer will automatically download all new subscribed episodes each time you start it up, but you must manually download older ones.
You can transfer podcasts from the computer to the iPad by syncing with iTunes, but must set it up in the right place. With your device selected in iTunes go to the Podcast page and set it however you want. But you have to sync each time you want new episodes to transfer.
Audio podcasts will show up in the iPod app. Video podcasts will show up in the Video app, though I've had them show up in the iPod app as well now and then.
I believe there are third party podcast apps, and they may automatically sync. However they will not work with iTunes. They will have their own library of podcasts.