  This work is devoted to study the global existence of strong and classical
solutions to compressible Navier-Stokes equations with or without density jump
on the moving boundary for spherically symmetric motion. We establish a unified
method to track the propagation of regularity of strong and classical solutions
which works for the cases when density connects to vacuum continuously and with
a jump simultaneously. The result we obtain is able to deal with both strong
solutions with physical vacuum for which the sound speed is $1/2$-H\"older
continuous across the boundary, and classical solutions with physical vacuum
when $ 1 < \gamma < 3 $. In contrast to the previous results of global weak
solutions, we track the regularity globally-in-time up to the symmetric center
and the moving boundary. In particular, the free boundary can be traced.
