A lot of cell phone stores will do this for you when you purchase a phone from them for a small charge, or sometimes free. This can be done on-line at: www.movemycontacts.com, worked like a charm for...
If your phone plan includes mobile to mobile calling it does.If you're using prepaid it generally won't unless your on the $1 a day mobile to mobile plan.
If the Motorola you have is unlocked to work on any network then yes it will, if it is not unlocked, then you will need to get in contact with the original network provider and ask them to unlock it