In this paper, we construct a new family of distance-optimal binary cyclic codes with the minimum distance $6$ and a new family of distance-optimal quaternary cyclic codes with the minimum distance $4$. We also construct several families of cyclic and negacyclic codes over ${\bf F}_2$, ${\bf F}_3$, ${\bf F}_4$, ${\bf F}_5$, ${\bf F}_7$ and ${\bf F}_9$ with good parameters $n,\,k,\,d$, such that the maximal possible minimum distance $d_{max}$ of a linear $[n, k]_q$ code is at most $d_{max} \leq d+8$. The first codes in these families have optimal or best known minimum distances. $145$ optimal or best known codes are constructed as cyclic codes, negacyclic codes, their shortening codes and punctured codes. All optimal or best known codes constructed in this paper are not equivalent to the presently best known codes. Several infinite families of negacyclic $[n,\frac{n+1}{2}, d]_q$ codes or $[n, \frac{n}{2}, d]_q$ codes, such that their minimum distances satisfy $d\approx O(\frac{n}{\log_q n})$, are also constructed. These are first several families of such negacyclic codes.