Notes and contacts not updating on Macbook Pro (10.9) from iPhone

Backing up your device does not effect syncing with iCloud. If you created notes in a non-iCloud account on your phone, for example, you would have to open each note and copy and paste the text into new notes created in the iCloud account in order to get them into iCloud. If you have contacts created in a non-iCloud account, you can either recreate the contact in the iCloud account, or if you have quite a few I can give you a process to copy them to iCloud without having to re-enter them if you're interested.

After copying the text from each note into new notes created in the iCloud account you will have duplicate notes on your device. When you're done, open the notes in the non-iCloud account and delete them to remove the duplicates.

For contacts, what account(s) are the contacts in now that you need to move to iCloud (On My iPhone, Gmail, Yahoo, an exchange account, etc)?

You can tell what account they are in by opening Contacts, tapping Groups, then selecting your non-iCloud groups one at a time to see the contacts that belong to each non-iCloud account.

Let's assume that they are scattered across multiple iCloud accounts as you stated. To get them all in iCloud, do the following:

1. Go to Settings>iCloud, turn Contacts to Off, choose Delete from My iPhone when prompted (they will still be in iCloud).
2. Download the app My Contacts Backup to your phone. (You'll need to get the "Pro" version if you have more than 500 contacts that you need to copy to iCloud.) Use this to backup the remaining contacts on your phone (from the other accounts) as a vCard attachment to an email that you send to yourself. Confirm that you have received the email.
3. Go to Settings>iCloud and turn Contacts back to On.
4. Go to Settings>Mail,Contacts,Calendars>Default Account (in the Contacts section), set this to iCloud.
5. Go to Settings>Mail,Contacts,Calendars...tap each of the other accounts that you are syncing contacts with (such as Gmail) and turn off contacts syncing. This will prevent having duplicates when you re-import them from the backup your made earlier.
6. Open the My Contacts Backup email and tap the attachment to import the contacts from the other accounts back to your phone. They should be in iCloud, and as a result, will sync to your other devices.

You're welcome.