When I search the web, all I get is hits on how to transfer your "contacts". "SIM" is an acronym for "Subscriber Identity Module". So, as I understand it, there is more information on that little thing than just my "contacts". I just got a new phone (with a new SIM card), and I want to move every bit from my old SIM to the new SIM, not just the "contacts". How do I do that?
Cloning a SIM card is not what you think it is. Apart from the simple contacts information, the only thing stored on it is a code which identifies your phone to your carrier. It's like a user ID, which the carrier can use to look up which plan you're on and what phone number you should have.
Although cloning a SIM card is possible, it's not something you want to do as a customer. It's a technique used by crackers to impersonate another user to the phone network, in order to illicitly receive their phone calls. It may be illegal where you live.
Think of the SIM card like a credit card. It belongs to the carrier: they just lend it to you for the purpose of using their service. If your bank sends you a new card, or you change bank, you don't "clone" your old card onto your new one: you just use the new card. And, just like a bank card, if your carrier didn't send you the new card as a replacement for the old one, then it refers to a completely different account.
If you simply want to move your old phone number to your new phone, you need to contact your (old) carrier to do this.