Backup stored procedure for full, differential and log backup
There are quite a few solutions to have your daily backups (full, diff and logs). You could:


Write your own script: it might be useful if you need a fine control on it, or if you need to respect some naming convention. I did not test it, but your code looks reasonnable.
Use the maintenance plans: very easy to set up, but less control. This might be a good solution in your case.
Use a 3rd party script / tool: Ola Hallengren's scripts are very good, but you can find plenty of others.