Are you taking the
SIM from the old phone and putting it into the new phone?
If so, all you need to do is copy all the phone numbers (contacts) to the
SIM card and put the
SIM in the new phone. Then you can just use the numbers off the
SIM or copy them to the new phone's memory. (this is what I did with my Nokia and Motorola phones).
As to your second question, yes, you can connect the motorola to the computer. Depending on which Motorola phone you have, you can connect using Bluetooth, USB data cable, WiFi or Infrared.