How should I run cables through interior wall to garage?
It depends what you mean by "cables". As long as it is a data cord (not power) and the opening is fire sealed, it is fine. What I do is use a cutoff saw (hack saw is fine) to cut a length of PVC pipe. Use a hole saw to cut through the drywall. Then, caulk the PVC pipe to the dry wall. It makes a perfectly clean tunnel through the wall. You can feed data cables through this tunnel. There is no code violation for doing this as long as the cables are data cables and you fill the tunnel afterwards with fire proof sealant.

Note that you cannot have any "wall penetration" between a garage and a bedroom, but it appears from your diagram that the adjacent room is not a bedroom.