Can I view the list of songs within a podcast while playing it?
There is a (non-obvious) way to see whatever information was provided by the person who made the podcast. Whether it lists the songs will depend on how much info they actually provided.

To see the info about a podcast, when listening to it, click on the main area of the display - as marked with a large red rectangle below:



This takes you to a sort of detailed view:



Clicking on the text description here takes you back to the first view (pictorial representation of the podcast).

One extra thing: Clicking on the little icon at top-right, that looks like a sort of list (shown in the small red rectangle in the first image above), takes you to the list of "episodes" that you have for this podcast, e.g.:





Edit: What about viewing podcast info on iOS 5 iPad?

This info subsequently moved to @JasonSalaz's separate iPad question