You can debug the script with -x, which will echo the commands and arguments.
bash -x script.sh
You can do this for specific portions of the file, too (from section 2.3.2 linked above):
set -x # activate debugging from here
ssh user@example.com ...
set +x # stop debugging from here
The output from -x might be a bit too verbose for what you’re looking for, though, since it’s meant for debugging (and not logging).
You might be better off just writing out your own echo statements – that’d give you full control over the output.