Excited nuclear matter at high temperature and density results in the creation of a new state of matter called Quark Gluon Plasma (QGP). It is believed that the Universe was in the QGP state a few millionths of a second after the Big Bang. A QGP can be experimentally created for a very brief time by colliding heavy nuclei, such as gold, at ultra-relativistic energies. The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory consists of two circular rings, 3.8 km in circumference, which can accelerate heavy nuclei in two counter-rotating beams to nearly the speed of light (up to 100 GeV per beam). STAR (Solenoidal Tracker At RHIC) is one of two large detectors at the RHIC facility, and was constructed and is operated by a large international collaboration made up of more than 500 scientists from 56 institutions in 12 countries. STAR has been taking data from heavy ion collisions since the year 2000. An important component of the physics effort of the STAR collaboration is the Beam Energy Scan (BES), designed to study the properties of the Quantum Chromodynamics (QCD) phase diagram in the regions where a first-order phase transition and a critical point may exist. Phase-I of the BES program took data in 2010, 2011 and 2014, using Au+Au collisions at a center-of-mass energy per nucleon pair of 7.7, 11.5, 14.5, 19.6, 27 and 39 GeV. It is by now considered a well-established fact that the QGP phase exists. However, all evidence so far indicates that there is a smooth crossover when normal hadronic matter becomes QGP and vice versa in collisions at the top energy of RHIC (and likewise at the Large Hadron Collider at the CERN laboratory in Switzerland). At these very high energies, the net density of baryons like nucleons is quite low, since there are almost equal abundances of baryons and antibaryons. It is known that net-baryon compression increases as the beam energy is lowered below a few tens of GeV. Of course, if the beam energy is too low, then the QGP phase cannot be produced at all, so it has been proposed that there is an optimum beam energy, so far unknown, where phenomena like a first-order phase transition and a critical point might be observed. On the other hand, there also exists the possibility that a smooth crossover to QGP occurs throughout the applicable region of the QCD phase diagram. Experiments are needed to resolve these questions. In this dissertation, I focus on one of the main goals of the BES program, which is to search for a possible first-order phase transition from hadronic matter to QGP and back again, using measurements of azimuthal anisotropy. The momentum-space azimuthal anisotropy of the final-state particles from collisions can be expressed in Fourier harmonics. The first harmonic coefficient is called directed flow, and reflects the strength of the collective sideward motion, relative to the beam direction, of the particles. Models tell us that directed flow is imparted during the very early stage of a collision and is not much altered during subsequent stages of the collision. Thus directed flow can provide information about the early stages when the QGP phase exists for a short time. A subset of hydrodynamic and nuclear transport model calculations with the assumption of a first-order phase transition show a prominent dip in the directed flow versus beam energy. I present directed flow and its slope with respect to rapidity, for identified particle types, namely lambda, anti-lambda and kaons as a function of beam energy for central, intermediate and peripheral collisions. The production threshold of neutral strange particles requires them to be created earlier, and these particles have relatively long mean free path. Thus these particles may probe the QGP at earlier times. In addition, new Lambda measurements can provide more insight about baryon number transported to the midrapidity region by stopping process of the nuclear collision. It is noteworthy that net-baryon density (equivalent to baryon chemical potential) depends not only on beam energy but also on collision centrality. The centrality dependence of directed flow and its slope are also studied for all BES energies for nine identified particle types, lambda, anti-lambda, neutral kaons, charged kaons, protons, anti-protons, and charged pions. These detailed results for many particle species, where both centrality and beam energy are varied over a wide range, strongly constrain models. The measurements summarized above pave the way for a new round of model refinements and subsequent comparisons with data. If the latter does not lead to a clear conclusion, the BES Phase-II program will take data in 2019 and 2020 with an upgraded STAR detector with wider acceptance, greatly improved statistics, and will extend measurements to new energy points.