This paper provides an analytical framework with foundations in stochastic geometry to characterize the spatio-temporal interference correlation as well as the joint coverage probability at two spatial locations in a cellular network. In particular, modeling the locations of cellular base stations (BSs) as a Poisson Point Process (PPP), we study interference correlation at two spatial locations $\ell_1$ and $\ell_2$ separated by a distance $v$, when the user follows \emph{closest BS association policy} at both spatial locations and moves from $\ell_1$ to $\ell_2$. With this user displacement, two scenarios can occur: i) the user is handed off to a new serving BS at $\ell_2$, or ii) no handoff occurs and the user is served by the same BS at both locations. After providing intermediate results such as probability of handoff and distance distributions of the serving BS at the two user locations, we use them to derive exact expressions for spatio-temporal interference correlation coefficient and joint coverage probability for any distance separation $v$. We also study two different handoff strategies: i) \emph{handoff skipping}, and ii) \emph{conventional handoffs}, and derive the expressions of joint coverage probability for both strategies. The exact analysis is not straightforward and involves a careful treatment of the neighborhood of the two spatial locations and the resulting handoff scenarios. To provide analytical insights, we also provide easy-to-use expressions for two special cases: i) static user ($v =0$) and ii) highly mobile user ($v \rightarrow \infty)$. As expected, our analysis shows that the interference correlation and joint coverage probability decrease with increasing $v$, with $v \rightarrow \infty$ corresponding to a completely uncorrelated scenario.