Cell phones run on two different types of networks: GSM and CDMA. These network types themselves are subdivided into various bands (e.g. GSM 850 MHz). Most recent phones will support a wide range of bands within one network type. GSM is predominant in most countries except the US, which uses both.
To verify that your current phone will work in another country, you first need to find out which bands your phone works on. This will be listed on the technical specifications, which you may be able to find online (if you can't, try calling your carrier). You'll then need to check which bands are used by carriers in the country you're going to. Wikipedia is good for this—for example, here's the list for Europe.
Data basically works the same way, although it is a bit more complicated. Basically, there are a number of different network technologies: HSDPA, HSUPA, HSPA, HSPA+, LTE (I'm not sure how complete this list is). The best way to do this is to check Wikipedia and work backwards (if their network supports, say, 1800 MHz LTE, you'll want to check if your phone does).
In addition, if your phone has not been unlocked, you will be forced to keep using your current SIM. This seems to be what you're planning to do, but I just want to emphasize that switching to a local SIM will not work, so if your carrier is willing to unlock your phone, do get that done.
The other thing to make sure to do is check with your carrier is that you do have international roaming enabled. If not, your phone basically won't work outside of the US, and it's a lot easier to get that enabled before you travel instead of trying to dial your carrier from abroad.