Lipid droplets (LDs) are neutral lipid storage organelles surrounded by a phospholipid (PL) monolayer. LD biogenesis from the endoplasmic reticulum is driven by phase separation of neutral lipids, overcoming surface tension and membrane deformation. However, the core biophysics of the initial steps of LD formation remains relatively poorly understood. Here, we use a tunable, phenomenological coarse-grained model to study triacylglycerol (TG) nucleation in a bilayer membrane. We show that PL rigidity has a strong influence on TG lensing and membrane remodeling: when membrane rigidity increases, TG clusters remain more planar with high anisotropy but a minor degree of phase nucleation. This finding is confirmed by advanced sampling simulations that calculate nucleation free energy as a function of the degree of nucleation and anisotropy. We also show that asymmetric tension, controlled by the number of PL molecules on each membrane leaflet, determines the budding direction. A TG lens buds in the direction of the monolayer containing excess PL molecules to allow for better PL coverage of TG, consistent with the reported experiments. Finally, two governing mechanisms of the LD growth, Ostwald ripening and merging, are observed. Taken together, this study characterizes the interplay between two thermodynamic quantities during the initial LD phases, the TG bulk free energy and membrane remodeling energy.