Did you use a PIN to protect the data on the SIM? If so, getting the data from it is much harder.
I've used the
Dekart SIM reader with good results.
Once you use the reader, getting the data off the SIM is quite easy, if it's not PIN protected. The reader comes with SIM card reader software that will grab all the info for you, including SMS data. It's pretty standard for digital forensics.
If it is PIN protected, the software will let you enter the PIN. If you don't know it, things get trickier, but it's still not too hard. There are tools that will let you unblock the SIM included, and it comes with a guide to help you do it.