How would I create a list of sub-folders in a given folder without the full path?

The following in a bash script:

find /Volumes/SpeedyG -type d >> file.txt

...works well to list the folders in that path in a text file,

/Volumes/SpeedyG/folder1  
/Volumes/SpeedyG/folder2  
/Volumes/SpeedyG/folder2

but the results are the full path to the folder.

What if I just wanted the folder name without the full path?

folder1  
folder2  
folder3  

For the GNU implementation of find, you can do this using the formatted output action printf:

          %P     File's name with the name  of  the  starting-point  under
                 which it was found removed.

So

find /Volumes/SpeedyG -type d -printf '%P\n' >> file.txt

If you want to remove all leading directory components, you can use %f instead of %P. If you have zsh, you can do the same using recursive shell globbing and the :t (tail) qualifier:

print -rC1 /Volumes/SpeedyG/**/*(ND/:t)

in which case you don't need find at all. With any POSIX sh, you could always do

find dir -type d -exec sh -c '
  for f do printf "%s\n" "${f##*/}"; done
' find-sh {} +

tree -di /Volumes/SpeedyG will give you a list of only subfolders.

tree -dia /Volumes/SpeedyG will give you a list of only subfolders to include hidden directories.

tree -diL 1 /Volumes/SpeedyG will give you a list of only subfolders 1 level in.

note: if a directory is a symbolic link to another directory, then that line of output will show the link and path to the target directory...
You could eliminate it with a grep statement like tree -di /Volumes/SpeedyG | grep -v "\->"

GNU basename removes any leading directory components from a full pathname.

$ basename /Volumes/SpeedyG/folder1
folder1

To process multiple pathnames, use the --multiple option:

‘-a’
‘--multiple’
     Support more than one argument.  Treat every argument as a NAME.
     With this, an optional SUFFIX must be specified using the ‘-s’
     option.

$ basename -a /Volumes/SpeedyG/folder1 /Volumes/SpeedyG/folder2 /Volumes/SpeedyG/folder3
folder1
folder2
folder3

In a script, a more efficient way of stripping off the largest prefix is to resort to parameter expansion like so:

file="/Volumes/SpeedyG/folder1"
file="${file##/*/}"

You can utilize this in a for loop:

for i in /Volumes/SpeedyG/*
do
    printf "%s\n" "${i##/*/}"
done > $HOME/files.txt

$ cat ~/files.txt
folder1
folder2
folder3