If the SIMs are the same size, just put the old SIM in the new phone. (If they're not, there are adapters to make smaller SIMs fit phones that need larger ones, but most of them come from China, so it takes a few weeks, which wouldn't help you now - but for a buck including shipping, you should have a set anyway.)
If they're different sizes, you have to ask the carrier to do an equipment swap to the new phone. Then, if you change back, you'll have to ask them to do it again. (Depending on the carrier, their attitude will be anything from returning a smile to being very annoyed. I've only done it with AT&T recently, but they act like you didn't ask them to do anything - it's "sure, it'll take a couple of minutes".)