Port-au-Prince is the capital and largest city of Haiti.
Port-au-Prince airport (PAP) is served by several major airlines primarily American Airlines, United Airlines and Delta as well as smaller flights from the Dominican Republic, Cuba, and other spots in the Caribbean. Taxis from the airport to your destination in Port-au-Prince will be about $20 for standard fare. Try to bargain down to $15. Tap Taps going to all places past the airport will cost about 10 gourdes ($0.25) and a community-created route map can be found here: http://taptapmap.org
From Santo Domingo: Caribe Tours, Capitol Coach Line and Terra Bus each run very modern buses daily to Port-au-Prince, each of the 3 companies departing from their own station along Av. 27 de Febrero. Caribe runs to Pétionville (in the hills above Port-au-Prince) that leaves at 11:00 am. Most all tickets currently cost $40 one-way, plus serious tax/border fees of about $26 and 100 DR, depending on the direction. Unfortunately, Caribe Tours' bus drops you off in Pétionville after dark so make prior arrangements with a trustworthy person to meet you and transport you to your lodging.
Another, less expensive option, is to take a guagua (Dominican minibus) from Santo Domingo departing 4 blocks NW of Parque Enriquillo, just W of Av Duarte, from a small parking lot within the elevated expressways of Espresso 27 de Febrero). http://horariodebuses.com suggests guava buses leave every 45min, but this is not always the case. Price is 400 DR (about $10, 5+ h permits a quick rest/meal stop) and arrive in the border town of Jimani. From there it is a 4 km walk or a 50+ DR ride by motoconcho to the border post. The border is apparently open 08:00-18:00 (if it respects its times).
In the past it was very easy to cross the border without submitting to any immigration procedures on either side, and although probably illegal, saved a few dozen dollars on bribes and was much faster too. Things are changing: passport control is now generally required leaving the DR, not just entering the DR. Entering Haiti legally is quick: fill out the green form and pay whatever amount the official asks (around 100 DR). There are no ATMs at the border. Moneychangers give gourdes for Dominican and American currency. Rates are fair. Protect Haiti's small green card in your passport, allowing you to leave Haiti without risking a penalty.
There's usually plenty of local transportation from the border to Port-au-Prince. Crowded tap-taps and buses can take you to Croix-des-Bouquets for about 75 gourdes (1-2 h), from where it is another hour to Port-au-Prince (bus, 5+ gourdes per route, summary network map @ http://taptapmap.org). Road ranges from very bad to good, and is prone to flooding. Peruvian UN soldiers at the border have confirmed that the road to Port-au-Prince is safe to travel with no incidents of robbery or kidnappings, but definitely try to arrive in Port-au-Prince before dark.
Tap-taps run along prescribed routes throughout the city. Most routes cost 10 gourdes ($0.25), though to get across the city you may need to utilize multiple routes, each of which charges separately.
Taxis typically are about 500 gourdes and should be used only during daylight. After dark, prices rise substantially, and you are at substantially greater risk for being mugged.
There are at least two banks with ATMs: Scotiabank and Sogebank. The closest Scotiabank to downtown is at the intersection of Boulevard Jean-Jacques-Dessalines and Rue Pavée. Even the ATM is closed on Sundays. Banks here close very early, even on the weekdays.
Eating out in Port-au-Prince is surprisingly expensive. Even at modest restaurants a full plate of food will usually cost around 200 gourdes. A good amount of food from street vendors will even cost up to 100 gourdes.
There are grocery stores all over town at least two in the center of town, both located on Capois: the Big Star Market in the Champ-de-Mars area and the Primera Market nearby the Hotel Olafson.
There are no cheap places to stay, just less expensive choices.
You should not be outside on the streets after dark unless you are wandering around the busy Champs-de-Mars area. Many travelers and guide books rate Port-au-Prince as the most dangerous major city in the Caribbean in terms of crime and personal safety, even more so than, say, Kingston, Jamaica or Port-of-Spain, Trinidad and Tobago.