Are you sure you want to come in April? The weather doesn't become reliably good until May, which is when most cafes and restaurants begin to put out their sidewalk seating. May is also when most of the outdoor festivals/events begin to happen. It's like night and day between the cold months and the warm months here.
Anyway, your profile doesn't give much information about what you like to do. You also didn't give a price range on what you're willing to spend on food or accommodation. Hopefully my suggestions won't be too hit or miss.
ARTS: The must-sees are the ICA on the waterfront, the Museum of Fine Arts (the amazing Chihuly exhibition is showing now), and the Gardner Museum. You should also look at what's happening at MassArt and the School of the MFA. Sometimes they have interesting free shows that are open to the public.
FOOD: Eat like a local.
Breakfast/brunch: For dim sum go to Chow Chau City or China Pearl in Chinatown. You can get decent breakfasts/brunches at Brownstone, Geoffrey's, Mike's Diner, Francesca's, and the Trident Cafe.
Lunch: If you're in town on a weekday afternoon, have the prix fixe lunch (around $24) at the Top of the Hub. It's a 3 course meal with a great view of the city. Cheap lunches ($10 or less) can be found anywhere if you don't mind burritos, pizza, or burgers. If you want to get out of downtown, take the B train to Super 88 in Allston (an Asian grocery store and food court with lots of variety) or the Silver Line (Washington St branch) to Dudley Square for Haley House Bakery Cafe (fresh, healthy wraps and hot meals). You can also take the C train to Coolidge Corner for crepes and then wander around and eat at one of the Jewish or Russian owned places. There's a good Israeli place for falafel not far from the train, but I don't remember the name.
Lunch/dinner: Jamaica Plain is a great neighborhood for food. Take the #39 bus and go either to the intersection of South Huntington Ave and Centre St for El Oriental (Cuban), Canary Square, or Tres Gatos (tapas). Or, continue taking the #39 until you reach central JP. Once there, go to JP Licks (the BEST ice cream in the city), Same Old Place (pizza), Bukhara (Indian), Purple Cactus (burritos/wraps), or JP Seafood Cafe (Japanese/Korean).
Back downtown, go to Montien (Thai), Mentei (Japanese noodles, cash only), Pho Pasteur (Vietnamese), Casa Romero (Mexican), Zocolo (more expensive Mexican), Addis Red Sea (Ethopian), and Mike's (Greek) on Newbury St.
There's good Central and South American food in East Boston, but you need to speak Spanish well to ask directions and order your meals. Dorchester and Roxbury have good Soul and southern food, but it may not be worth your time during a weekend visit.
Dinner: You could eat at most of the lunch places above or spend a little more at The Beehive, Pops, Estragon (it's out of the way, but good tapas), Tremont 647, or Legal Seafood.
Drink: Milky Way, Canary Square, Flash's, Wally's, and Towne.
Dessert: JP Licks (ice cream) and Finale (baked sweets).
HISTORIC SITES: The Freedom Trail and Faneuil Hall Marketplace are two of the most touristy things you could possibly do. If you like Colonial history and Revolutionary War stuff, have at it. But, you could also just wander the city and stumble upon pieces of history everywhere you go. One of MLK's former residences is near the Mass Ave. subway station, JFK's birthplace isn't far from Coolidge Corner, Malcolm X worked at the Parker House, Ho Chi Minh was here for something, etc, etc.
SHOPPING: Newbury Street is where most people go. There are interesting places for furniture, design, and art elsewhere, but they're hidden and it would be hard to explain the directions. You should definitely go to Haymarket on Saturday to see one of Boston's oldest farmer's markets in operation.
GAY PLACES: Club Cafe (bar/dancing for the 20 somethings; tea dance on Sunday), Fritz ("sports" bar for the 40+ crowd), House of Blues (Sat is gay night), Paradise (fun sleezy dance club).
ACCOM: There are plenty of B&Bs and hotels.
FYI, the majority of my recommendations are in Boston. Cambridge has its own scene, with the most interesting areas being Central Square, Harvard Square, and Inman Square.
Have a good visit! If you have questions or need more info, inbox me.
P.S. It's Boston Common. There's no S at the end.