You can forward them to your email address; add the date/time and person or phone number as that info does not get forwarded.
BitPim is another option, but use it cautiously as it can also mess up your phone if you don't know what you are doing. Simply LOOKING at the information it provides will not harm anything, and if you just down,loading the text messages, it should be fine (IF you can get it to recognize and "see" your phone).
BitPim is a pretty awesome tool, and can do a lot, but it's when you start tinkering and adding or changing settings that things get messed up. Just looking and grabbing data will not harm your phone.
I'm not answering your question (sorry) but I have a similar question. Isn't there a way to download the texts onto my computer if I use a usb to sync it? I've tried and can't get that to work, but I feel like It should....