The time required to bring the turbo up to a speed where it can function effectively is called turbo lag. This is noticed as a hesitation in throttle response when coming off idle.This is symptomatic of the time taken for the exhaust system driving the turbine to come to high pressure and for the turbine rotor to overcome its rotational inertia and reach the speed necessary to supply boost pressure.
A Wastegate is simply a turbine bypass valve. It works by diverting some portion of the exhaust gas around, instead of through, the turbine. This limits the amount of power that the turbine can deliver to the compressor, thereby limiting the turbo speed and boost level that the compressor provides.
Engine calibration - fueling and ignition timing. Under boost, it is crucial that there is no engine-killing detonation occurring within the cylinder. This is done by fine tuning the air/fuel ratio a bit rich to help cool the combustion gas, and by tuning the ignition advance curve to ensure that combustion chamber pressures stay below the level that causes unburned fuel to ignite ahead of the advancing flame front.