If you’re thinking about studying abroad in Rome, I cannot stress enough how helpful it is to start learning Italian beforehand. Even if you only manage to take an intro class before you leave (like me), I promise it’s worth it. Just in case you don’t get a chance though, here are some basics.
First of all, learn your basic courtesies/greetings. That means “boun giorno” (“good morning/afternoon”), “buona sera” (“good evening”), “ciao” (“hi/bye”), “scusa” (“excuse me/sorry”), “grazie” (“thank you”), “prego” (“you’re welcome”), and “per favore” (“please”). I know that seems painfully obvious, but somehow it never seems to occur to me until I land in a new city and frantically realize that I know zero words in the local language. Do yourself a favor, and memorize that now.

The next important phrase is “non parlo italiano” (“I don’t speak Italian”). Having said that, a look of mild terror when anyone tries to speak to you in Italian will probably convey the same sentiment, so it’s really a matter of your preference. Relatedly, “parli inglese?” (“do you speak English?”) is a good one to know too. Personally I find it a little presumptuous to walk up to someone and launch right into English, just assuming that they’ll understand (even though they usually do), whereas prefacing whatever I need to say with a quick “parli inglese?” seems a little more respectful.
Once you’ve summoned up the courage to start venturing into shops, you’ll need a few more phrases in your repertoire. In a lot of stores the workers will immediately yell “prego” or “dimmi,” when you walk in, which is basically “what can I do for you?” Don’t get thrown off by the fact that this tends to sound more like a command than an offer. If you’re not quite ready to buy anything you can respond with “sto solo dando un’occhiata” (“I’m just looking”), or “quanto costa?” (“how much does this cost?”). Finally, “posso provare?” (“can I try some?”) is a good food-phrase to know, although it’s really only useful in gelato shops (a.k.a. I use this one pretty much everyday).

Next problem: you’ve successfully said something in Italian, but you have no idea what the response was. You have a few options here, starting with “poui ripetere, per favore?” (“can you repeat that, please?”). If that doesn’t work, follow with “poui parlare più lentamente, per favore?” (“can you speak slower, please?”). If you’re still lost, offer a sad “non capisco” (“I don’t understand”) and resort to your charades prowess.
Finally, a word of advice: read! If, like me, your Italian isn’t quite up to reading normal books (by my estimation, after two semesters of Italian, my ideal conversation partner is a two-year-old,) try a book that you’re really familiar with. I’m currently making my way through Harry Potter, and even though the language and grammar is more advanced than my actual comprehension level, I know the book well enough that I can follow.
Buona Fortuna! (Good luck!)