Saving SMS on Memory Card Sony Ericsson M600i

SMS are like a small email if you have uploaded it to your PC just open it
Depending on your phone model, you can transfer files wirelessly via Bluetooth" or Infrared (IR). You may also be able to transfer files to a PC via a USB cable. The USB cable is recommended when you are transferring large files or a large number of files.
It shoudl include some software
Do  you have any software that connects the PC to the mobile phone via usb cable?
http://www.sonyericsson.com/cws/products/accessories/overview/dcu-60?cc=US&lc=en

I dont have a Sony  rather a different mobile and it has a software called Join ME
Once my mobile is connected to the PC via Join Me
 I can type in text messages and send and receive from my desktop
To save the text part I open a message from my in box or sent box within join Me then copy it to a TXT

I have this cable but I cannot see the SMS messages or copy them.....

What's the model of your Sony?
http://www.gsmarena.com/sony_ericsson-phones-19.php
Could you provide more details

The Model is M600i
1) I need to move the SMS from the SIM card to the memory card.
Once sucessfully done I can copy them to the PC.

Ok right
Tools>File Manager>more>device>
or Tools>controlpanel>device>storage wizard
Be sure to format a blank memory card via the phone before use.
Filemanger>More>Format
I actually found the manual
http://www.sonyericsson.com/cws/download/1/198/359/1193000731/M600i__UG_R3a_EN.pdf
Go to page 54 and rename your memory card so you know what it is

Save this
Sony Ericsson Tips And Tricks
http://www.mobile9.com/invboard/lofiversion/index.php/t17065.html

I can see the memory card while I want to copy a picture from Phone Memory to external memory card.
But in case the SMS, I have only the option to Move to "SIM card" or a folder created on the phone only.

Use the file manager and move the SMS to a folder so you can manage it.
Please read the manual. Page 54 File manager that may help you.
Otherwise consider a USB cable I also get fed up with navigating around the phone memory cards sd card readers etc
 it's so much simpler from PC using a usb.
have you install the Sony Erricson Suite on the PC
Go to page 59- Synchronization and  Backup that may assist you.

The proble is ; that in the "File Manager" I cannot see the SMS folder. I only view (Picture, Music, Video, Document and Other).

File Manager doesn't work if you would like to copy your sms to pc. You can try FMA: http://fma.sourceforge.net/index2.htm (a program specially designed to copy sms from a Sony Ericsson mobile phone to your pc).

I used the software.
When I try to connect the M600i through the bluetooth, a message appear on the taskbar from the float's mobile agent: your device is not supported by current windows drivers, Install or update yor drivers and try again!

Any update?

I used the USB cable , but it doesn't work. I didn't know how to configure USB port.

Ok,  sounds like we hit a dead end again.
are you running Vista or XP?
USB in XP is automatically detected once the cable is plugged in and the mobile conected,
 a Folder should appear on teh desktop and also the mass storage device on the task bar,
just open that folder and you should see your stuff on the memory card not the Simcard.?
Ensure your saving the mesages to the memory stick not the sim card
In your message options, you can specify to have your SMS/MMS's saved onto memory stick instead of the phone's memory by going into
Options->Settings->Other->Memory in use, this should be standard for most mobile phones.
If that feature is not supported then use it with a filemanager such as Fexplorer you could manually move the mail folders back and forth from the memory stick to the sim card.
Then use the dual micro and an SD card reader
Running out of ideas here.