For groups: you can check out this recent paper of John Conway, Heiko Dietrich, and E.A. O'Brien (DOI) for results and conjectures on counting the number of groups of a given order (I also seem to remember a recent article of Conway's in the Notices of the AMS (or maybe the Bulletin) on this subject).

For fields: there is a unique isomorphism class of fields of size $p^n$ for each prime $p$ and each positive integer $n$, so one can figure out the asymptotic from the prime number theorem.

For rings: the OEIS has information on this sequence here.

For groups: you can check out this recent paper of John Conway, Heiko Dietrich, and E.A. O'Brien (DOI) for results and conjectures on counting the number of groups of a given order (I also seem to remember a recent article of Conway's in the Notices of the AMS (or maybe the Bulletin) on this subject).

For fields: there is a unique isomorphism class of fields of size $p^n$ for each prime $p$ and each positive integer $n$, so one can figure out the asymptotic from the prime number theorem.

For rings: the OEIS has information on this sequence here.