With the increasing share of renewable energy, integrated electricity-gas distribution systems (IEGDS) have developed rapidly, highlighting the necessity of their study. To address the challenges of multi-facility planning and reliability assessment in IEGDS with renewable energy integration, a distributionally robust optimization (DRO) planning model is proposed that incorporates explicit reliability constraints. Reliability is quantified using the system average interruption frequency index (SAIFI) and the expected energy not supplied (EENS), for which an explicit reliability evaluation model is provided. Using Wasserstein distance, a DRO planning model is employed to address the stochastic nature of renewable energy and gas-electric loads. Case studies show that increasing facility types not only reduces total costs but also enhances system reliability and flexibility. Uncertainties in renewable generation and electricity/gas demand can lead to over a 60% increase in total costs. Additionally, higher reliability requirements further increase costs, but the resulting planning solutions are more valuable and informative for practical implementation.