If you're a coffee drinker, get ready for some straight up espresso goodness. The coffee bars here don't substitue milk and don't do flavorings. This is where you need to get just an espresso, a cappuccino, or (my favorite) an espresso macchiato, which is an espresso with a spot of milk on top. If you'd like a pastry as well, "cornetto" is what they call a croissant. In my experience, coffee+pastry was never more than 5 euro (and it was usually no more than 2 or 3).
There are about a bazillian little pizza al taglio places in Rome (which are not the sit-down kind of pizzerias, though there are at least a bazillian of those there, too) that give you however much pizza you want (I've found that gesturing is a useful communication tool), and it's priced by weight. Pricing pizza by weight like this makes soooo much more sense than the by-the-slice way we do it here in the states, and it usually ends up cheaper than you'd expect it to. I highly recommend doing that for lunch or dinner at least half the time you're there (not all pizza places will be open late, but with Rome's lively bar scene, you're sure to find some).
Some places offer an item called "suppli" which are these ORGASMIC little balls of fried rice with mozzarella and a bit of tomato sauce inside, and they're usually 1 euro a piece. If you find some, try it, and love it, you'll need to take a trip to the Trastevere neighborhood, just outside of the main city center. There's a place just off Viale di Trastevere called Sisini Venanzio, though the nondescript little sign above the entrance just says "i suppli." Depending on the time you get there, there will be a line (more like a mob, actually) of Italians spilling out the doors. Almost the entire 3 months I was studying in Rome--and living around the corner, mind you--I walked passed this place every day without realizing what it was. My last day in the city, I finally noticed the sign, went in, and was blown away. They have all different kinds of suppli, and they're all huge, and they might be more than 1 euro each, but not more than 2. Even if you're not on a quest to find more suppli, I highly recommend venturing down to this neighborhood. The food is usually better and more affordable than the bulk of the fare you'll find next to major monuments and tourist attractions.
That said, my favorite little lunch place is in the Campo de' Fiori. During the day, the piazza is a wonderful outdoor market (with produce and other things), full of locals AND tourists (as opposed to just tourists). On the northwestern end of the campo, opposite the Cinema Farnese, there's an amazing little bakery (with Italian pastries, deli sandwiches, and pizza) called Forno. While I was in school over there (my school was housed in a building just off the Campo), I'd eat here often after class. At the time, I thought that the little store across the ally from it (on the left if you're facing the storefront) was also a part of it, but, looking at their website, I can't tell if that's actually the case. Anyway, right under the big "FORNO" sign is the entrance. You can order pizza al taglio at the counter directly on your right when you walk in, or you can go to the back, where I think they have sandwiches and pastries. My recommendation is to get some of the pizza rossa (that's pizza with just red sauce), because it's the best pizza rossa I've ever had in my life. They might ask you (or motion, if you're obviously a foreigner who doesn't speak English) if you want your designated portion folded into a sandwich, and you should say "Si!" because this is the way to eat it.
The storefront on the left (whether or not it's actually part of Forno) sells reasonably priced sandwiches (REAL Italian sandwiches--not these silly American "panini") and various pastries. If I remember correctly, they'll heat sandwiches up for you if you ask (though I definitely can't remember how to ask for that in italiano...). Many of the sandwiches are made with fresh mozzarella di bufala and various veggies. If you see one with a strange-looking floppy orange thing in the middle, get it. This is "fiore di zucca," literally "flower of squash," and it's very subtle in it's taste, but very popular amongst Italians. The sandwiches run a few euro each. Also, if you'd like something sweet to round out your lunch, this place makes a pastry filled with ricotta in one end and chocolate in the other. I'm pretty sure they're called "ravioli di ricotta" (don't be confused, though--it's a sweet pastry, not pasta). If you're lucky, you'll get one that's still warm, and you'll be able to experience the miracle of chocolate and ricotta cheese oozing together in the middle of a soft buttery pastry shell. They're about the size of a croissant, but not made with flakey pastry dough...it's some kind of soft, slightly breadier (but sweet!) pastry...
And as for gelato, you'll probably be good with whatever hole-in-the-wall place you find. If you want to do a gelato-splurge, Giolitti is the most well-known and the most extravagant, in my experience. And even for their ginormous 3-scoop white-chocolate-dipped cone with fresh panna (whipped cream) on top, it's less than 10 euro. Possibly less than 5, but don't quote me on that. Just please, pleeeeeaaaaase do not go to Blue Ice. It's a CHAIN store, and it's NOT a good one. The flavors are colored with insane amounts of dye to make them as eye-catching as possible for tourists. (And on that note, if you enjoy the more natural route of food-making, San Crispino is a good gelateria with less traditional flavors that are made with all-natural ingredients. I've only been to the one near the Trevi Fountain, but according to the website, there are other locations as well. It is a bit on the pricier side of gelato, though.)
And most importantly, do some exploring! You're bound to find some amazing little places for yourself, just by walking around aimlessly. Put that tourist map away for a little bit and let yourself get lost! ;)