Today have fired up my new Samsung Galaxy S II. To find that managing SMS messages is not as intuitive as on my old - non-smart Sony Ericsson. Cant seem to find a way to save messages to SIM - or anywhere else - for example. And - I'm not up to android progamming!
I don't know if there is a way to save them directly on the device, but I do know of a tool called SMS Backup + that I have been using almost since my fist android phone in 2008, and before I used SMS Backup +, I used SMS Backup.
This application will backup your SMS messages and Call Log to your gmail. It puts them in to SMS/CallLog labels so they don't clutter your inbox, but it is a very handy thing to have all your messages backed up. Especially when your wife says "I told you to get 3 loafs of bread", you can pull up and say, "Nope, see you said 30 loafs"...
SMS Backup + can not only backup your SMS/Call Log, but it can also restore them. The developer says MMS is also "coming soon". (but he has said that for a while not, not that I am in any rush for that feature). And, did I mention, the app is free.
Even if I was able to save them to my phone, I would have deleted them 100x over by now with all the times I have done factory resets, new sdcards, etc. Gmail also has way more storage available then my SIM, device, or sdcard. It is also more reliable. I have never lost data stored in my gmail, but I have had 4 sdcards fail on me.
For just backing up and restoring messages, you can try SMS Backup & Restore which backs up to an XML file on your SD card. You can use the XML file to view messages in a web browser or MS Excel, etc. as well.