Listen or Watch in iTunes or on an iPod
Of course, you can listen to podcasts directly from iTunes on a computer. However, you probably want to involve mobile devices so students don't tie-up a computer to listen. iTunes works best with iPods for listening to podcasts because you just plug in the iPod and podcasts can be automatically synchronized onto the device. The podcasts are neatly organized on the iPod. You can even set iTunes to automatically delete older episodes.

It's a good idea to organize podcasts into playlists. You can create playlists in iTunes by clicking the + button at the bottom-left of the window. Name the playlist and then drag and drop episodes onto it.  You can arrange the episodes and even rename them to make it clear to students which episode they are should listen to or view. Playlists created in iTunes are transferred to iPods when synced.

Listen on Other Devices
Many devices can play audio and video.  However, only iPods, iPhones, and iPads are designed to sync with iTunes. You can still transfer podcast episodes to those devices from iTunes, but you'll have to do it manually.

If your device has a memory card attach a card reader to your computer. Insert an expansion card that fits your device. Open the expansion card. Then open iTunes' Podcasts window. Position your iTunes and memory cards' windows so you can drag and drop individual episodes from iTunes and onto the card's window. 

Once copied to the card, the card can be inserted into a handheld and played. Note that some devices have their own podcast software like the Zune Marketplace for Zune players and Google Listen for Android devices. If you are using a Palm OS handheld read Using your Palm OS Handheld as an MP3 Player from K12 Handhelds.