Multiferroicity can be induced in strontium titanate by applying biaxial strain. Using optical second harmonic generation, we report a transition from $4/mmm$ to the ferroelectric $mm2$ phase, followed by a transition to a ferroelastic-ferroelectric $mm2$ phase in a strontium titanate thin film. Piezoelectric force microscopy is used to study ferroelectric domain switching. Second harmonic generation, combined with phase-field modeling, is used to reveal the mechanism of coupled ferroelectric-ferroelastic domain wall motion. These studies have relevance to multiferroics with coupled polar and axial phenomena.