Quantum and nonlinear optics has garnered a lot of interest in the last few decades due to its applications in a plethora of fields such as sensing, spectroscopy, frequency metrology and quantum information processing. Optical microresonators have compactified the realization of several nonlinear phenomena since their inception, besides enabling fundamentally new paradigms in light-matter interaction. Planar microcavities such as ring resonators are particularly interesting for their ease of fabrication in a massively parallel fashion and their small mode volumes. Additionally, advances in fabrication techniques have enabled ultra-low losses and ultra-high quality factors in these planar CMOS compatible materials such as silica and silicon nitride. In this dissertation, we harness the ultra-low loss of silicon nitride microrings along with its high third-order Kerr nonlinearity for two applications previously unexplored in this platform: bright squeezed light generation and dual comb spectroscopy. We begin by examining the background in each of these fields before detailing our approach to realizing a compact, robust and scalable alternative to the existing state-of-the-art. The first part of this dissertation investigates squeezed light, a quantum state of light that is promising for quantum enhanced sensing and quantum computation or communication in the continuous variable regime. Nearly all previous chip-based sources of squeezing have been rather large (of the order of many wavelengths in height and width), and work based on the second-order nonlinearity, which is found only in restrictive crystals lacking inversion symmetry. Here we report the first realization of squeezing using an integrated parametric oscillator above threshold based on the ubiquitous third-order nonlinearity. The on-chip microrings enable the generation of broadband continuous wave squeezing in contrast to the narrowband squeezing produced by their macroscopic counterparts, and this broad bandwidth is essential for high-bit-rate quantum communications. Furthermore, we show continuous electrical tuning of the degree of squeezing using integrated platinum microheaters incorporated above coupled microring resonators. We conclude our exploration of quantum photonics in silicon nitride with a path towards deterministic entanglement generation. In the second part of this dissertation, we look into a classical application of the same platform: dual comb spectroscopy (DCS). This technique builds upon an actively pursued aspect of silicon nitride rings -- that of frequency comb generation. Again, using the third-order Kerr nonlinearity, a series of narrow, equidistant lines can be generated in the frequency domain, to form a spectrum that is called a frequency comb. Two such combs with slightly different line spacings can be used for dual comb spectroscopy, which is an emerging technique for fast acquisition of broadband optical spectra. Since DCS does not involve any moving mechanical parts that are used in conventional spectrometers, it is much more robust. We generate, for the first time, two frequency combs on the same chip using a single laser, and use it for broadband spectroscopy of dichloromethane. We demonstrate simultaneous soliton modelocking of both combs and observe long mutual coherence times of 100 \textmu s. Our results pave the way for an on-chip, fully integrated dual comb spectrometer.