You probably should stop listening to the first friend when it comes to car stuff.
1st of all a set of all-season (AKA no-season) tires have no place on a "high performance car". 2nd of all, ALL tires need to be balanced, and that goes double for a "high performance car". Imagine a Ferrari at 190mph with out of balance tires, it would take less than 2 minutes before things started breaking.
As far as the 2nd friend, toe can make the car twitchy, but it won't make the car vibrate. As mentioned by Gville re-balance the tires.