There is no such thing as transferring the program. It must be installed on the new computer using your Legacy CD or downloaded install file. The Legacy website has a very good list of how-to's and FAQs. Check this out: http://www.legacyfamilytree.com/helpMovingBetweenComputers.asp
Yes, generally speaking, since Legacy Family Tree resides on your present computer with a link to Legacy Family Tree.com and the software program that created it, you should be able to see it on any other computer. You can download the program again on your new computer or use the online download web site, and all your information should appear. You can always call Legacy Family Tree, and someone will get back with you to help on the phone.