To begin, you will need to visit Google.com/settings.
You will want to back up your account data before you delete the account. To do that, click on "Data liberation." On the next screen click on "Download your data."
By clicking on the create archive button, Google will prepare a ZIP file containing all of your data for you to download. Once the archive is built, you'll be able to download the ZIP file.
Now that you have your data stored somewhere safe you can delete your Google account. Go back to the Google.com/settings page we started on earlier and scroll to the bottom of the page. There you will find a link to close your entire account.
If you are a Google Apps user you will notice the link isn't to close your account, it's to delete your Google profile and Google+ features. Closing your account can only be done by your Google Apps admin. You can go ahead and delete your profile information, but will need to contact your admin to actually delete your account.
Once you have clicked on the link to close your account, you will have to acknowledge each Google service you no longer want to use, enter your password, and check a few more boxes confirming you truly want to delete your Google account.
Once you confirm everything and click on the delete button, there is no going back. Your account will no longer exist. With your data backed up and stored on your computer you can sort through what information you would like to import into another service, as well as what information you would like to discard.