You say, "... through an insulated wall." When going horizontally THROUGH a wall ('front to rear') with wire, run your wire right up against the 'side' of the stud AND directly behind the outlet, to the other side. This way, all your wiring (for future reference, and for 'others' later on), are all run where the outlet is. At the same time, the wiring doesn't 'interfere' with the insulation since the wire is kinda sandwiched between the side of the stud and the insulation.
If, on the other hand, you want to run your wire through an insulated wall ('left to right'), usually, the wiring is run UPWARDS, directly from the room below. Sure, you can run it horizontally from stud to stud but that's done in very RARE cases. In a rare case, the electrician locates the stud and then uses a LOOOOONG drill bit and drills a hole in the drywall and RIGHT throuh the stud, all in one shot. In other words, an 'access' hole isn't drilled first, whereafter, you'd then drill into the stud. It's somewhat of a pain to run the wiring because you have to pull ALL of it through as you go from stud to stud. You really can't just fish it through and pull because the holes in the stud aren't exactly perpendicular to the stud itself. The hold is on an angle towards the other side of the wall. See what I mean? If you can avoid it, don't do it this way.
At worst, when running wire from below, you need to cut out a 'knotch' of drywall just above the baseplate, and drill your hole to the room below. After fishing, you patch the hole. You know, there are some decent books on DIY Electrical Work. Before you go crazy, why don't you spend some of the $$$ you saved in doing the work yourself by INVESTING in a good book. Try to find one that 'concentrates' on renovation and not just new construction.