Take a snapshot of your current Hass.io config and pull it down from your Pi based server. Shutdown the PI and keep the SD card intact incase anything doesn’t go right.
Install the linux distribution of your choice (a debian based distro seems to be the go to with few issues reported)
Install Hassio using the directions below.
Install the samba add on and drop the snapshot file you saved previously into the backup folder.
Restart HAss.io. Restore using the snapshot.
Most components should come back without issue. Some will require some tweaking if they use hardware attached to the machine USB device paths etc.
Don’t forget that things are likely pointing to your Pi IP currently. You either need to update things to the new server IP or have the new server take the old Pi IP.