Android - How to use ADB to list remote directories?

Easy to do – as in any Linux system:

1. Connect your device as usual
2. log into it using adb shell
3. use the ls command to list files in the current directory, and the cd command to change directories (i.e. walk the tree / navigate through directories)

For details on the commands, check for the corresponding Linux man pages – e.g. man ls and man cd.

Edit: If you're "afraid" of the command line, you can also use an ADB file manager (see here for examples).

For rooted device (having the su binary):

adb shell su -c ls -R *

This will list all folders with their contents and their subfolders.