How do I capture the output from the ls or find command to store all file names in an array?

To answer your exact question, use the following:

arr=( $(find /path/to/toplevel/dir -type f) )

Example

$ find . -type f
./test1.txt
./test2.txt
./test3.txt
$ arr=( $(find . -type f) )
$ echo ${#arr[@]}
3
$ echo ${arr[@]}
./test1.txt ./test2.txt ./test3.txt
$ echo ${arr[0]}
./test1.txt

However, if you just want to process files one at a time, you can either use find‘s -exec option if the script is somewhat simple, or you can do a loop over what find returns like so:

while IFS= read -r -d $'\0' file; do
  # stuff with "$file" here
done < <(find /path/to/toplevel/dir -type f -print0)