iTunes will backup everything on the iPhone, including SMS messages and contacts.
In iTunes, with the iPhone selected on the left, click the Info tab and make sure Contact syncing is enabled.
Besides,there is a article about How to transfer SMS from iPhone to computer, which can help you transfer SMS from iPhone.
Hope it worked for you so much.

i think they all save when you do a backup. when my Iphone was exchanged from apple i installed my backup i had all my  SMS messages back.

Question how can you access the saved SMS on iTunes?


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I know we can transfer iPhone Contacts, Notes, Calendar, Music, Photos and Videos to my iPhone 3GS.But how can we backup our previous iPhone 3G SMS to iPhone 3GS?

First,if you jailbreaked your phone than it’s damn easy just SSH your iPhone 3G and Navigate to /private/var/mobile/Library/SMS and Save sms.db on your computer then follow the vice versa with iPhone 3GS just copy sms.db from your computer to iPhone 3GS /private/var/mobile/Library/SMS folder.

Here is the simple to follow steps to Restore SMS History from Previous iPhone to New iPhone. All you have to do is to copy SMS backup files from old iPhone back up folder to new iPhone back up folder.

Navigate to the iTunes backup folder
1.Windows User : C >> Users >> User Name >> AppData >> Roaming >> Apple >> Computer >> MobileSync >> Backup >>
2.Mac OS X User : ~/Libary » Applications Support » MobileSync » Backup.

In the backup folder there will be X number of folders according to number of phones you synced to your iTunes. For example i have two iPhone 2G and 3G so i have two folders named with 40 random hex characters. But i need to copy SMS from iPhone 3G so how do i identify?
To identify your particular iPhone backup sync both phones with 5 minutes of interval and note the time in your computer clock while syncing.
Now Sort the view by date–this will let you easily tell which subfolder belongs to which phone.
Now you knows the folder where your previous iPhone 3G backup is saved, so explore to that particular folder (iPhone 3G) and copy these two files to iPhone 3GS folder. Make sure you must synced new iPhone 3GS at least once so back up folder will be created.
3d0d7e5fb2ce288813306e4d4636395e047a3d28.mddata
3d0d7e5fb2ce288813306e4d4636395e047a3d28.mdinfo

I think you can't do it on a new iPhone. You have to put them on your itunes. But if your iPhone allows installing third party software (when you have done a so called 'Jailbreak'), you may download a piece of software called iSIM. This will allow you to batch copy all your phone numbers to the iPhone without the need for a PC or Mac.
What's more, some related toturials about iphone Contact to computer.
For Mac:how to backup iPhone SMS/Contact to Mac

For PC: How to backup iPhone SMS/Contact to computer
Hopefully these work for all of you like it just worked for me!