Cool places ~ it's hard to say because everyone ideas of cool are a little different. That being said...
Places to Eat:
Abe & Louies, an upscale steak house
Sanctuary: a tapas place down on the waterfront
Parrish Cafe: great sandwiches
Atlantic Fish Company: good seafood
the Barking Crab: also good seafood
anywhere in the North End (it's NEVER called Little Italy by locals): delicious Italian food; follow your nose and stop anywhere that catches your fancy
Mike's Pastry: *the* pastry shop in Boston, also in the North End
Coolest Bars (by which I think you mean clubs):
Ned Devine's/Parris: Faneiul Hall
Saint: in the Back Bay
The Good Life: kinda between the theatre district and Downtown Crossing
The Roxy: close to New England Medical Center
Pool Halls:
The Rack: Faneiul Hall
Big City: Allston/Brighton
Malls: is there such a thing as a cool mall? Copley Place has a mall, as does the Prudential Center and there's the Cambridgeside Galleria. They are...well...malls.
Cool Places in General:
again, this is going to depend on your personal tastes. the Commons and the Gardens are beautiful. Fenway Park is a must for any sports fan. The Freedom Trail and Duck Tours are good for the general stuff. Newbury Street has great shops and restaurants. History is EVERYWHERE. The Constitution, the aquarium, Beacon Hill is beautiful. Charles Street is definitely worth a wander and a stop at Cafe Vanille for coffee or tea and a treat. LOTS of museums, harbor cruises...really, there's too much to list.
As everyone has already said, it's a very expensive town. Saving money is tough. Expect to drop some cash. If you are really limited then make sure to do your research and choose wisely. It's easy to drop $100 bucks just on dinner OR cover and drinks, let alone a night that includes both.
You should be able to get contact info on all these places through google or the front desk at your hotel. It's a great town. Enjoy yourself!!!