I'm trying to make a simple shell script to backup files on the same machine but I keep running into errors I don't know how to resolve

Ok, so I use an Ubuntu server 18.04 for my small business and I have very minimal knowledge outside of basic security and running what I need to run. So, apologies if this is a really silly issue.

I want to set up a simple backup script that backups files and stores them in another directory on the same machine.

To do this, I followed this example by Ubuntu: https://ubuntu.com/server/docs/backups-shell-scripts

I created the relevant directories and updated the script to this:

#!/bin/bash
####################################
#
# Backup to NFS mount script.
#
####################################

# What to backup. 
backup_files="/home/server/testserver/logs"

# Where to backup to.
dest="/home/server/backups"

# Create archive filename.
day=$(date +%A)
hostname=$(hostname -s)
archive_file="$hostname-$day.tgz"

# Print start status message.
echo "Backing up $backup_files to $dest/$archive_file"
date
echo

# Backup the files using tar.
tar czf $dest/$archive_file $backup_files

# Print end status message.
echo
echo "Backup finished"
date

# Long listing of files in $dest to check file sizes.
ls -lh $dest

I saved the bash inside "/home/server/backups" and then tried to run it on some test logs. However, I get this error:

-bash: ./backup.sh: /bin/bash^M: bad interpreter: No such file or directory

I researched this online and tried running the file as "bash backup.sh" and removing the "#!/bin/bash" line altogether, both resulted in this:

backup.sh: line 7: $'\r': command not found
backup.sh: line 10: $'\r': command not found
backup.sh: line 13: $'\r': command not found
backup.sh: line 18: $'\r': command not found
.tgzday2427erver/backupstestserver/logs
backup.sh: line 21: $'date\r': command not found
backup.sh: line 22: $'echo\r': command not found
backup.sh: line 23: $'\r': command not found
tar: Removing leading `/' from member names
tar: /home/server/testserver/logs\r\r: Cannot stat: No such file or directory
tar (child): /home/server/backups\r/ns31202427\r-Friday\r.tgz\r: Cannot open: No such file or directory
tar (child): Error is not recoverable: exiting now
tar: Child returned status 2
tar: Error is not recoverable: exiting now
backup.sh: line 26: $'\r': command not found
backup.sh: line 28: $'echo\r': command not found
Backup finished
backup.sh: line 30: $'date\r': command not found
backup.sh: line 31: $'\r': command not found
ls: cannot access '/home/server/backups'$'\r': No such file or directory

So can anyone explain what is happening here and help me resolve the issue so I can safely backup my server? :)