When do I spin up my rocket's second stage in KSP?

You don't.

While spinning your rocket during ascent - or indeed at any point during its flight - does have a stabilization effect, this also makes the rocket harder to control because the controls will be spinning along. Meaning if you try to tilt your rocket in one direction, you have to account for the spin, and good luck with that. During ascent, you want to keep control over your rocket to perform your gravity turn, so spinning is a bad idea. If you want to stabilize your rocket during ascent, you could try adding wings to the base of your rocket, or larger reaction wheels if the ones in your control point are not enough.

Also, a solid fuel second stage? Solid rocket boosters cannot be throttled down or shut down once ignited, and provide excellent thrust to weight ratio but poor efficiency, which makes them useful when trying to lift a heavy first stage, not when you need to control a spacecraft that's about to get into orbit.