Kathy,
A SIM card is a Subscriber Identity Module, which are used in GSM phones. These contain the subscribers account information as well as the phone number. Your Verizon phone uses CDMA technology, and therefore won't have a SIM card.
There's no "relatively cheap" way to make calls from Europe to the U.S. The rates seem to be close, regardless of which cell network is used. My cell network charges $1/minute for calls in country and $2/minute for calls back to North America. Texts are 60¢ each.
Your easiest option would probably be to obtain the "travel phone" from Verizon, but try to limit voice calls. If you use text messaging, the costs should be quite manageable. However, be sure to review the roaming rates so that you're very clear on what your costs will be.
Happy travels!