Over the last two decades, the field of quantum computation has grown into a major field of physics and the trapped ion systems are one of the most promising candidates for the development of a quantum computer. As high-fidelity gate operations with few ions have already been demonstrated, the experimental challenges for quantum information processing lie in transferring these techniques onto scalable quantum systems. The presented work describes the development of a scalable experiment for trapped ion quantum computation with 40Ca+ and 88Sr+ ions. The incorporated ion trap is a planar, segmented Paul trap which enables reconfiguration of the stored ions in flexible quantum information registers to scale up quantum computation. In order to decrease the heating rate and the collision rate with background gas, the trap is operated in a cryogenic environment. The walls of the cryostat are used as magnetic shields by exploiting skin-effect to enhance the coherence times of the qubits. Measurements resulted in an attenuation of 120 dB for the magnetic shielding for 50 Hz signals. Vibration isolation was integrated in the design of the cryostat so that the positions of the ions in the trap are static with respect to the optical setup, resulting in a well-defined phase of the interaction field with respect to the qubit transition. In addition to the cryogenic apparatus, optical setups and electronic control systems for the operation with 40Ca+ and 88Sr+ ions were installed, which allowed a characterization of the entire system using trapped ions. The measurement of the coherence time of 40Ca+ showed a Gaussian decay of the Ramsey contrast with a 1/e-time of 18.2(8) ms and the obtained heating rate was 2.14(16) phonons/s.