Os x mountain lion 10.8.4

SOURCE: How do I sync Notes with my Mac laptop and my iPhone using Mountain Lion?

Hi Rachel,

In order to sync your Notes between your Apple devices, please use the following steps:

• Go to your iCloud settings and sign up for an @me.com email address.
  
• Once you have an @me.com email address, write a note in the Notes app on your iPhone.
  
• Go to the iCloud website and open up the "Mail" application
  
• Underneath all of your email folders (Inbox, Sent, etc.) there should be a folder called "Notes"
  
• Open that folder and read your note!

SOURCE: How do I use iCloud on the new OS X Mountain Lion?

All you do is go into "System Preferences" and then click on "iCloud". From there you will be able to activate your account and proceed.

Please note that you will have to set up an @me.com email address from Apple. Without this you will not be able to sync between devices like you wish to do.

SOURCE: How much does Mountain Lion cost to download?

You can download Mountain Lion by going to the Apple Store. It costs $20 to get.

However, if you purchased your new computer after June 11, 2012 (which your question seems to indicate you did) then it will not cost you anything and will be completely free.

Go to this webpage and enter in your details to get your free copy.