This thesis details the construction and operation of a new quantum simulation experiment for studying bosonic (and in the future fermionic) atoms in interesting triangular superlattice structures. We show that this machine can reliably and stably produce samples of ultracold potassium and rubidium in the lowest band of the kagome lattice. We discuss the characterisation and validation of the apparatus as well as preliminary results towards probing the flat energy band of the kagome lattice, a key experimental goal of the group. A major obstacle to faithfully implementing the kagome lattice, as well as other superlattices, is the stringent control of the optical phase of the lattice laser beams. We discuss the mechanisms which have been developed to achieve the required control and stability; and demonstrate that our machine has sufficient control to allow us to deterministically prepare and probe a range of interesting lattice types. In addition to superlattice structures, our apparatus also allows us to prepare monochromatic triangular and honeycomb optical lattices and preliminary experiments in these lattice geometries are discussed. The use of potassium in our experiment affords us the ability to tune atom interactions by means of a Feshbach resonance. We demonstrate control of tunnelling and interactions in the many-body system to observe the Mott-insulator transition in the triangular and kagome lattices. We also detail current progress towards achieving a state in thermodynamic equilibrium in the kagome flat band and demonstrate the creation of a negative absolute temperature state for potassium atoms in the triangular lattice. In the future we hope to realise a quantum gas microscope capable of single-site imaging of the individual atoms in the superlattice. We discuss progress towards this goal including the characterisation of an in-vacuum objective, as well as proposed super-resolution techniques which harness excited state light shifts to partition the lattice.