So, what everyone else said - especially Zokuku.
First you need to decide if where you are going has a CDMA network. If it does, then you need to choose whether you want to make calls on Verizon's network partner at Verizon's international rates, or whether you want to use a local prepaid SIM.
The advantage of making a call on Verizon's international roaming system is that your number will work overseas - so if someone in the US wants to call you, it will ring through. The disadvantage is that it will cost a lot of money. Where "a lot" is normally $0.50 - $2 per minute. And data is usually completely outrageous - and by "outrageous" I mean like $0.05/kB, which means that loading Engadget.com's webpage would cost about $25.
The alternative is to buy a prepaid SIM in which case you will get a local number and it will be a lot cheaper - particularly data. This is your only option if the country doesn't have a CDMA network that Verizon supports.
So if you go the Verizon international roaming route, you need to call Verizon and enable international roaming, and while you are there ask about their international calling plans. Usually for something like $5/month you can save a lot of money, and then call them afterwards and disable it when you get back.
If you want to go the prepaid SIM route, you need to call Verizon and make sure that your phone is unlocked for GSM networks.
I really like accessing data while I'm travelling because internationally, free WiFi access seems much harder to find than in the US, and I don't like doing things like checking my bank balance in an internet cafe. So because I use a lot of data, I always go with the prepaid SIM if I can.