The security requirements for future 6G mobile networks are anticipated to be significantly more complex and demanding than those of 5G. This increase stems from several factors: the proliferation of massive machine-type communications will dramatically increase the density of devices competing for network access; secure ultra-reliable low-latency communication will impose stringent requirements on security, latency, and reliability; and the widespread deployment of small cells and non-terrestrial networks, including satellite mega-constellations, will result in more frequent handovers. This paper provides a set of security recommendations for 6G networks, with a particular focus on access and handover procedures, which often lack encryption and integrity protection, making them more vulnerable to exploitation. Since 6G is expected to be a backward-compatible extension of 5G, and given that secure systems cannot be effectively designed without a clear understanding of their goals, it is imperative to first evaluate the limitations of the current generation. To this end, the paper begins by reviewing existing 5G access and authentication mechanisms, highlighting several critical vulnerabilities in these procedures. It then examines potential 6G challenges and concludes with actionable recommendations to enhance the security, resilience, and robustness of 6G access and handover mechanisms.