The Eighth Doctor audio story “Situation Vacant” is streaming for free on BBC Radio 4 Extra On Demand, which is available to everyone internationally, unlike BBC iPlayer. It’s a two-part, 60 minute long story that follows the Doctor’s participation in a companion-choosing contest. It’s a lot of fun and I think it’d make a good introduction story, especially if you don’t follow the rest of the arc to its conclusion (because the arc gets into stuff that you will not be aware of).
I highly recommend checking it out.
It’s available for 7 days, after which the next story, Nevermore, will be available to listen. Please check this out, I swear you won’t regret it.