Samuel,
I am closer to a solution. I figured out that by default the backup for SQL Server appends data instead of deletes data. I fixed that part by deleting the backup file before running the backup script. I still am having trouble as I can manually run the .bat file, but it seems that the task scheduler will not. I will take a closer look at the link that you provided to see if that will help out.
This is what my batch file looks like now:
cd /d "C:\Program Files\Microsoft SQL Server\MSSQL10_50.SQLEXPRESS\MSSQL\Backup"
del MyBackupFile.bak
"C:\Program Files\Microsoft SQL Server\100\Tools\Binn\SQLCMD.EXE" -i"C:\SQLBackupScripts\MySQLFile.sql" -S "localhost\sqlexpress"
cd /d "C:\Program Files\Microsoft SQL Server\MSSQL10_50.SQLEXPRESS\MSSQL\Backup"
copy MyBackupFile.bak \\AnotherServer\AShareOnTheServer$\MyBackupFile.bak /y
cd /d "C:\SQLBackupScripts"
The SQL file has not really changed:
BACKUP DATABASE [MyDatabase] TO DISK = N'C:\Program Files\Microsoft SQL Server\MSSQL10_50.SQLEXPRESS\MSSQL\Backup\MyBackupFile.bak' WITH NOFORMAT, NOINIT, NAME = N'MyDatabase-Full Database Backup', SKIP, NOREWIND, NOUNLOAD, STATS = 10
GO
So once again, the above code works fine now when I manually run the backup script. Now I need to figure out how to automate the backups.