We study a system of neutral atoms trapped in a three-dimensional optical lattice suitable for the encoding, initialization and manipulation of atomic qubits. The qubits are manipulated by applied electromagnetic fields interacting with dipole moments of the atoms via light shifts, Raman transitions, Zeeman shifts, and microwave transitions. Our lattice is formed by three orthogonal one-dimensional lattices, which have different frequencies so that interference terms average to zero. This geometry allows considerable freedom in designing the component one-dimensional lattices, so that they provide not only confinement but also independent control in each dimension. Our atomic qubits are initialized from a laser-cooled atomic sample by Raman sideband cooling in individual lattice potential wells. We have demonstrated accurate and robust one-qubit manipulation using resonant microwave fields. In practice such control operations are always subject to errors, in our case spatial inhomogeneities in the microwave Rabi frequency and the light shifted qubit transition frequency. Observation of qubit dynamics in near real time allows us to minimize these inhomogeneities, and therefore optimize qubit logic gates. For qubits in the lattice, we infer a fidelity of 0.990(3) for a single pi-pulse. We have also explored the use of NMR-type pulse techniques in order to further reduce the effect of errors and thus improve gate robustness in the atom/lattice system. Our schemes for two-qubit quantum logic operations are based on controlled collisional interactions. We have experimented with two schemes in order to probe these collisions. The first involves manipulation of the center-of-mass wavepackets of two qubits in a geometry corresponding to two partially overlapping Mach-Zender interferometers. Unfortunately, this scheme has proven extremely sensitive to phase errors, as the wavepackets are moved by the optical lattice. The other scheme starts with two qubits in spatially separated traps, and utilizes microwaves to drive one or both qubits into a third trap in-between the two qubits. Once the wavepackets overlap, the collisions create a large energy shift which can be probed spectroscopically.