We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lovász from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs contain a Hamiltonian path, and that all sufficiently large such graphs even contain a Hamiltonian cycle. The third conjecture, due to Smith from 1984, states that for $r\ge 2$ in every $r$-connected graph any two longest cycles intersect in at least $r$ vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph which can be used to improve the best known bounds towards all the aforementioned conjectures: First, we show that every connected vertex-transitive graph on $n\geq 3$ vertices contains a cycle (and hence path) of length at least $Ω(n^{13/21})$, improving on $Ω(n^{3/5})$ from [DeVos, \emph{arXiv:2302:04255}, 2023]. Second, we show that in every $r$-connected graph with $r\geq 2$, any two longest cycles meet in at least $Ω(r^{5/8})$ vertices, improving on $Ω(r^{3/5})$ from [Chen, Faudree and Gould, \emph{J. Combin. Theory, Ser.~ B}, 1998]. Our proof combines combinatorial arguments, computer-search and linear programming.