If you are just looking for somewhere to chill with a really good cup of coffee and some wi-fi, you can head to:
Murky in Eastern Market, small place so seating may be tricky, I think they have wireless there but you may want to check. They make fabulous espresso drinks using counter-culture coffee!
Tryst in Adams Morgan, lounge atmosphere, great food, counter culture coffee also, free wi-fi on weekdays only.
Mayorga in Columbia Heights, local coffee roaster, really nice ambiance, wi-fi every day, really delicious and string espresso from their own label. One of my faves!
Busboys and Poets in the U St Area, great space, ambiance, wi-fi everyday, coffee is not my fave but it is fair trade. Bookstore, lounge, restaurant and performance space, one of a kind!
Baked and Wired in Georgetown, just down the street from Barnes and Noble, not sure about wi-fi but they make really fabulous espresso drinks there, using counter culture coffee of course! They recently expanded so they have seating available and are now open on weekends as well.
Open City in Woodley Park, sister of Tryst, also with wireless access and the same counter culture brand coffee, but less ambiance there.
Circa in Dupont, wi-fi and even french press coffee if you like. Not sure about hours for wireless.
In Chinatown, I am sad to say that there are no real coffee shops, though there is now a super cool place you can go for a stunning atmosphere, coffee and free wireless... the new glass enclosed courtyard at the art museum! Though plugging your laptop in there may prove challenging, it is certainly worth checking out.
Update: new spot I just discovered, the "zen lounge" juice bar, cafe and espresso bar at Mint Fitness at the bottom of Adams Morgan one block north of 18th and U St. Comfy space with free wireless. You don't need to be a member to head to the cafe, though you'll likely want to purchase something.
If you don't have a laptop, here are some suggestions:
Free, short term, Kramerbooks in Dupont on Conn Ave at Q St. MLK Library or any public library, free. By GWU, At 21st and Penn, there is a store and cafe called Pangea and they have some internet stations with free access in their cafe. Check for store hours before heading there.
For paid internet access, there are stations at the International Youth Hostel on 11th between NY and Mass, though you may need to show ID at the front desk. In Cleveland Park, Foster Brothers Coffee right next to the metro has stations you can pay to use, also in Dupont, the Newsroom has stations you can pay to use as well, they are on Conn at Florida.
That covers a good section of town for now. See the previous wi-fi posts for links with complete lists!