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Over the past few decades, rapid development of laser cooling techniques and narrow-linewidth lasers have allowed atom-based quantum clocks to achieve unprecedented precision. Techniques originally developed for atomic clocks can be extended to ultracold molecules, with applications ranging from quantum-state-controlled ultracold chemistry to searches for new physics. Because of the richness of molecular structure, quantum metrology based on molecules provides possibilities for testing physics that is beyond the scope of traditional atomic clocks. This thesis presents the work performed to establish a state-of-the-art quantum clock based on ultracold molecules. The molecular clock is based on a frequency difference between two vibrational levels in the electronic ground state of 88Sr2 diatomic molecules. Such a clock allows us test molecular QED, improve constraints on nanometer-scale gravity, and potentially provide a model-independent test of temporal variations of the proton-electron mass ratio. Trap-insensitive spectroscopy is crucial for extending coherent molecule-light interactions and achieving a high quality factor Q. We have demonstrated a magic wavelength technique for molecules by manipulating the optical lattice frequency near narrow polarizability resonances. This general technique allows us to increase the coherence time to tens of ms, an improvement of a factor of several thousand, and to narrow the linewidth of a 25 THz vibrational transition initially to 30 Hz. This width corresponds to the quality factor Q = 8 × 10^11. Besides the molecular quantum metrology, investigations of novel phenomena in state-selected photodissociation are also described in this thesis, including magnetic-field control of photodissociation and observation of the crossover from ultracold to quasiclassical chemistry.