Saving texts from old phone
Yes the app automatically sends all the text messages you receive to Treasuremytext.com. You don't have to do anything. It will give you a little notification each time it saves a message so you'll know it's working.
If you delete messages from your phone then they will still be there on Treasuremytext. So you can keep your phones memory free of old or private messages and in case you lost your phone you've still got them all on Treasuremytext.
It doesn't cost you anything to use the app. Messages are saved from your G1 using your Data connection (the same one used for internet browsing). This means there are no SMS charges and with the G1 you probably have an unlimited Data tariff so there are no costs.