Splintered state is a pretty good run I think to start them out with.
It does quite a few things:
Introduces paranoia to the group.
Introduces the fact that you shouldn't try to shoot your way out of every scene (IE, Chimera encounter)
Gives them a chance to start making contacts (my team did Splintered state on the 2nd or 3rd run, and got introduced to quite a few big wigs that they have been schmoozing with ever since, treating them well)
Lots of legwork
Moral choices (take the big bucks, or side with the good guys)
Introduces plot twists (like the entire run(s).
And if they go along with the second part, it can set up an interesting sub plot (for example, a sub plot I'm running throughout the entire campaign is Brackhaven's upcoming impeachment proceedings). And that lets me do all kinds of side runs. For example, I had Dana Oaks kidnapped, and sent me team looking all over for them). I've had my team blackmailed for a gruesome murder that may or may not have been related to the impeachment.
Another fairly simple run that is a bit more straight forward is this:
Johnson hired a courier to transport and item, courier was bribed and then killed upon handing the item over. Johnson wants the team item back.
Scene 1 Johnson intro, team needs to get to the crime scene and deal with KE (Johnson can pull some strings if the team seems a bit reluctant to talk to them to get them through the crime scene).
They can get various security footage from the surrounding buildings, they can search the crime scene (being smart), Sweet talk the officers (great way to get a low end KE officer as a contact).
Eventually they found an Urban Brawl championship ring from a local team. This led them to digging into the fact that only 3 of the team is left, the rest have died.
This leads them to the guy on the video that stole the item for whatever nefarious purposes.
It wasn't overly complicated, it can get new players accustomed to role playing their characters, there is time for the face to shine, time for the decker to shine (getting the security camera feeds), time for some knowledge skills to work out (ie knowledge of the urban brawler league), and some combat with the guy and his henchmen when encountered. I kept a lot of the encounters in fairly nice neighborhoods so the teams had to be careful and couldn't go all pink mohawk without attracting a ton of attention, so they kept the strike surgical.