Since they both use micro SIMs there should be no problem. However, if there is, head for the carrier's store with the S4 and ask them if they can do an equipment swap for you. (I'd be amazed if they charged you anything. AT&T gave me a free SIM when I needed to use an old feature phone for a few days while my new phone was in for replacement.)
If you're planning on buying a new S4, of course you get a new SIM with it.
If you're worrying about the contacts stored on the SIM, they'll copy them over in about a minute. (They should be on Google anyway, not on the SIM.)