Utilizing small-signal direct modulation and relative intensity noise measurements, we investigate changes in the modulation response, the differential gain partial derivative g/partial derivative n, the nonlinear gain coefficient epsilon, and the damping factor K, which result from the following three structural modifications to GaAs-based multiple quantum well lasers: 1) the addition of strain in the quantum wells; 2) an increase in the number of quantum wells; and 3) the addition of p-doping in the quantum wells. These modifications are assessed in terms of their potential for reducing the drive current required to achieve a given modulation bandwidth, for increasing the maximum intrinsic modulation bandwidth of the laser, and for improving the prospects for monolithic laser/transistor integration. The differential gain is increased both by replacing unstrained GaAs-Al0.25Ga0.75As QW's with strained In0.35Ga0.65As-GaAs QW's and by increasing the number of strained QW's, ultimately leading to substantial improvements in modulation bandwidth at a given drive current. However, in both cases, the increased differential gain is offset by corresponding increases in the nonlinear gain coefficient, leading to relatively constant values of K and hence little variation in the maximum intrinsic modulation bandwidth. By further adding p-doping to the In0.35Ga0.65As-GaAs MQW active region, we have been able to simultaneously increase partial derivative g/partial derivative n and decrease K, yielding very efficient high-speed modulation (20 GHz at a dc bias current of 50 mA) and the first semiconductor lasers to achieve a direct modulation bandwidth of 30 GHz under dc bias (heat-sink temperature = 25-degrees-C). Since our laser structures show no significant carrier transport limitations, the measured K factor for the p-doped devices implies a maximum intrinsic 3 dB modulation bandwidth of 63 GHz.