Computing the geometric Picard lattice of a K3 surface involves two complementary tasks: ruling out classes that cannot occur and constructing enough divisor classes to generate the lattice. This minicourse will present approaches to both tasks. The first part focuses on how the action of Frobenius constrains the specialization of the Picard lattice and provides upper bounds on the geometric Picard rank. We will explore several approaches, including reductions modulo p and searching for p-adic obstructions to lifting divisor classes to characteristic zero. The second part concerns ongoing work with Emre Can Sertöz focused on quartic K3 surfaces, which generates lower bounds on the geometric Picard rank. Starting from numerical period approximations, we identify putative divisor classes expected to be represented by smooth rational curves, and explain how to reconstruct and rigorously certify exact equations for the corresponding curves. These curves generate a saturated Galois-stable sublattice of the geometric Picard lattice; when its rank matches the upper bound obtained in the first part, we recover the full geometric Picard lattice as a Galois module.