The first discovery of an extra-solar planet (exoplanet) around a main-sequence star, 51 Peg b, discovered using Doppler spectroscopy, opened up the field of exoplanets. For more than a decade, the dominant way for finding exoplanets was using precise Doppler spectroscopy to measure the radial velocity (RV) changes of stars. Today, precise Doppler spectroscopy is still crucial for the discovery and characterization of exoplanets, and it has a great chance for finding the first rocky exoplanet in the Habitable Zone of its host star. However, such endeavor requires an exquisite precision of 10-50 cm/s while the current state of the art is 1 m/s. This thesis set out to improve the RV precision of two precise Doppler spectrometers on two 10-meter class telescopes: HET/HRS and Keck/HIRES. Both of these spectrometers use iodine cells as their wavelength calibration sources, and their spectral data are being analyzed via forward modeling to estimate stellar RVs. Neither HET/HRS or Keck/HIRES deliver an RV precision at the photon-limited level, meaning that there are additional RV systematic errors caused by instrumental changes or errors in the data analysis. HET/HRS has an RV precision of 3-5 m/s, while Keck/HIRES has about 1-2 m/s. I have found that the leading cause behind HET/HRS’s “under-performance” in comparison to Keck/HIRES is temperature changes of the iodine gas cell (and thus an inaccurate iodine reference spectrum). Another reason is the insufficient modeling of the HET/HRS instrumental profile. While Keck/HIRES does not suffer from these problems, it also has several RV systematic error sources of considerable sizes. The work in this thesis has revealed that the errors in Keck/HIRES’s stellar reference spectrum add about 1 m/s to the error budget and are the major drivers behind the spurious RV signal at the period of a sidereal year and its harmonics. Telluric contamination and errors caused by the spectral fitting algorithm also contribute on the level of 20-50 cm/s. The strategies proposed and tested in this thesis will improve the RV precision of HET/HRS and Keck/HIRES, including their decade worth of archival data. This thesis also documents my work on characterizing exoplanet orbits using RV data and the discovery of HD 37605c. It concludes with a summary of major findings and an outline of future plans to use future precise Doppler spectrometers to move towards the goal of 10 cm/s and detecting Earth 2.0.