The One-Stop DC Trip Planning Website
If there is one website you go to to plan your trip to DC, it should be www.washington.org. They have pages listing things to do by season, 100 Free and Almost Free Things To Do, and every possible attraction there is to see in the greater DC metro area.
Monuments, The National Mall, & The National Cherry Blossom Festival
The Phool Muddhi will be during one of DC's most festive and beautiful events, The National Cherry Blossom Festival, make sure not to miss the walk around the Tidal Basin (near Jefferson Memorial) and to of course visit all the national memorials on the National Mall.
http://www.nationalcherryblossomfestival.org
Smithsonian
The Smithsonian operates 19 museums and galleries and the National Zoo, and they are all free. Some of the most popular are the Air & Space Museum, which has everything from Apollo 11 to military drones, the the American History Museum where you can see Kermit the Frog and Julia Child's Kitchen, and the Museum of Natural History, where you can see a giant squid that looks like an alien from Independence Day, or if that grosses you out, the Hope Diamond.
Other Museums
There are a lot of popular new museums popping up all over DC that aren't a part of the Smithsonian. There is an entry fee for each (usually around $20). The most popular are the Spy Museum, the Crime Museum, and the Newseum.