Your biggest worry should be
the fact that the 60 series tires have a much lower load capacity than 70 series of the same width, so unless you're running around 36 lbs with a light load, you're overloading them, and you dare not haul a full load in the car. You also have a small change in your overall gear ratioes, so that your engine is turning perhaps 5 to 7 per cent faster for the same actual road speed, although your speedometer will be reading faster by the same percentage, too, and you'll be winding up miles on the odometer faster than you're really driving them. Of course, for a young person, having a speedometer that reads higher than reality can be a major safety factor.
The lower profile tire also changes the relationship between the centerpoint of the steering axis and the point at which the center of the tire meets the road slightly, so you may experience a little better road feel and response than with stock tires, or it just may turn out to be an excess of kickback and a tendency to follow grooves in the road.
I do think a step in the opposite direction from stock, to 205/70, is a good idea for anyone who carries serious loads, but I prefer higher performance tires, so I'm running 225/60 X 14, which may require thin spacers on some models of rims, etc.
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