I wanted to upgrade my phone from the Samsung Core to the S4 or S5, get a 24 month plan but keep my current SIM card. Is this possible?
If you're worried about transferring contacts/information, SIM cards only store contacts and nothing else. They're only used for your cell service. You can easily transfer contacts by saving them to your Gmail and then logging into the new phone or just send them over through Bluetooth. I stopped saving my contacts to my SIM cards 3 years ago and it's much more convenient having them on Google now.
I would recommend you get a new SIM card, you should have the option of getting an NFC enabled SIM card at no extra charge when buying a new phone, as a SIM card is included with the phone. These are neat because they allow you to make mobile payments and other cool stuff.