I would highly reccomend a stroll along Via Vincenzo Lancia to see the remains of the Lancia Factory and Palazzo Lancia - not too far away is Ex stabilimento Lancia which is the original Lancia factory which is on the south western corner of Corsa Racconigi and Corso Peschiera.
There is the former FIAT factory at Lingotto, but you need to visit the art gallery on a day it is open to get onto the roof. There is an excellent shopping centre full of restaurants on lower floors.
For a total Beta pilgramage - visit Chivasso - not much to see other than what remains of the factory behind Centro Commerciale, Chivasso in the service road - a rather decayed modernist style factory. The Town Centre and Piazza Della Republica in the middle is very charming and you can almost imagine that some of the older folk you'll see around could have been responsible for building your Beta!
The ultimate Lancia pilgramge has to be Fobello which is the birth place of Vincenzo Lancia. I have a crazy idea to go there in my Coupe when all is well with it again!