Depends on what you want to use the phone for, how much you want to use it.
Sign onto AT&T's international plans if your phone is locked to AT&T. If you've had the phone for awhile, you may be eligible to have it unlocked so you could use local carrier SIMs in the countries you visit.
Main advantage of getting an AT&T international package (you can turn it on and off for just a month) is that you keep your regular phone number, in case someone needs to reach you while you're abroad and you don't have to deal with figuring out how to get local SIMs and so forth.
Main disadvantage is that it's costly, for voice, data and text, and you can't use much data because it's way more expensive to roam.
If you have an unlocked phone, you have more options. One would be to use the T-Mobile Simple Choice for a month. Then you can roam and get data in other countries for free but the speed is throttled to about .1 Mbps. Compared to LTE in the US, you could be talking 100 times or slower but it's probably good enough for light surfing and things like Google Maps can work.
If you want a lot of fast data, then you get a local SIM from a carrier in that country. Prices vary by country and carrier and what kind of package you want. It's not bad if you just want enough data to use some apps. occasionally.
If you want a lot of data, like say 2 GB or more, like you get in the US per month, then expect to spend 20-40 Euros.
With a local SIM, you have a local phone number, so your American friends won't know the number to reach you and if they dial it, they're dialing international long distance.
A lot of people just go for maximal data and use things like iMessage to text with their other iPhone-owning friends or use FaceTime or Skype so that they don't have to worry about voice minutes on the foreign mobile network, just how much bucket of data you have.
I've bought as much as 10 GB of mobile data, for a week or two week trip, mainly because a lot of smaller hotels in Europe have unreliable and slow Wifi.