For use in Europe, you'll want a GSM cell phone with the following characteristics:

1. Unlocked, unless you use international roaming with your US-based carrier. Most cell phones sold through cell phone companies like AT&T or T-Mobile will be locked to be able to use SIM cards only from that carrier. Some places like cellhut.com offer unlocked cell phones.

2. Has the 900 and 1800 GSM band capability. Quad-band phones will have those bands, plus the 1900 and 850 GSM bands used in the US. Tri-band phones sold in the US typically have 850/1800/1900, which may give spotty coverage (900 is more common than 1800 in most parts of the world). Tri-band phones sold in 900/1800 markets typically have 900/1800/1900, which covers both of the needed bands, plus the more common US band (which should give decent coverage in urban areas of the US). Some cell phone vendors in the US do offer 900/1800/1900 tri-band phones. Dual-band phones sold in the US are generally 850/1900.

Verizon uses CDMA, so Verizon cell phones that do not also have GSM capability will not be useful in Europe.

With a suitable phone, you can buy a prepaid SIM card appropriate to the countries that you are visiting (usually once you are there, or you can buy them before going at a higher price). Or you can use international roaming with a US-based GSM carrier like AT&T or T-Mobile.