To make Sarah Jane and Rose continue laughing the make-up artist painted a moustache on David Tennant without them knowing until he came on set. (Which is why at one point you see Rose point at him while laughing.)
Elisabeth Sladen warned the crew about the difficulties with working with the K-9 prop, which had a reputation for breaking down. Despite several years of improvement in technology, Sladen's warnings were proved correct.
The Doctor's reaction when talking to Sarah Jane for the first time required little acting for David Tennant. Having already been a fan of Elisabeth Sladen since childhood, he was genuinely thrilled to work with her.
Sarah's last line to K9 is "We've got work to do." This was also the last line of the original series run, at the conclusion of the Seventh Doctor story Survival.
The K9 prop used at the end of the episode was a refurbished version of the original radio-controlled unit built for The Invisible Enemy, now in the possession of Mat Irvine.