I have a Verizon Galaxy S3 currently and I'm on a plan through my relatives.
Can I just buy a new smartphone specific for Verizon, like this Galaxy S5, and put my 4G LTE SIM card into it to start using it instead?
Yes, if you purchase a phone that is VzW compatible and the phones use the same size SIM card (such as the S3 and S5), you can just move your SIM card to the new phone.
you can also move an sd card, if compatible with the new phone.