As others have said, if you like good food, you MUST go to the North End.
If you want to eat cheap, for $1.60 you can get a slice of Sicilian pizza at Parziale's bakery on Prince Street (down just over a block from Hanover St). Ideally go there at 9:30 AM. The pizza is coming out of the oven for the first of several times during the day. You can get it all day but shortly after it comes out of the oven it is the best pizza anywhere. Did I say it's only $1.60 a slice.
Ask for Aldo, or Augie. Those are the son / father team who own the bakery and do the baking. They are about fourth generation. Pretty sure Parziale's is the oldest bakery in Boston, probably New England. It's a gem that most don't know about. Tell them Jim sent you. They are great Patriot fans and my friends.
After you have eaten the pizza, you can go back to Hanover Street. Take a right onto Hanover from Prince and a couple of doors down you will run into Mike's Pastry. I think a couple of other guys have already mentioned Mike's. Mike passed away a couple of years ago but his son carries on his legacy of the best pastry anywhere. The first cannoli I ever had was at Mike's. The shells are made right there and filled with a delicious fresh mix of ricotta and powdered sugar. They also have yellow cream filled. And can dip in chocolate or pistacios or several other variations. All the Italian pastries are fresh and superb. I also love the rum cakes, and paraginas, and Neopolitans. But my favorite is their eclairs. Fresh, delicious shells, filled with the richest and best yellow cream you will EVER eat and topped with the richest and best chocolate frosting you will EVER find on an eclair. Best eclairs anywhere. Get ready to spend 4 or 5 bucks for each pastry. Pricey treat. But worth every penny. I spring for a whole bunch for all special occasions and celebrations, including my Super Bowl party every year.
If you want a breakfast pastry. Go up Hanover to Modern Pastry. Their coffee rolls and pastry twists are tremendous. And only $1.50 ea. Gotta get there before 9:00 AM though or they may be out.
On another night, go back to the North End for dinner. You're gonna have to open your wallet a little wider but the Italian food of the North End is hard to beat ANYWHERE. So many good restaurants I don't know where to start to recommend. Virtually all have menus posted for viewing from the street. Maybe you could try my friend Donato Frattoroli's new restraunt "Il Molina" on Commercial St across from Union Wharf between the North End and the waterfront. He has opened several restaraunts over the past 30 years (all still operating). I saw him last week having coffee in Modern Pastry. He said that this is his latest. Think it is scheduled to open this week. He is a fabulous cook and host. Maybe you could try the place and let me know how it is. I'm sure you won't be disappointed.
Buon appetito.