Bucharest is linked through direct daily trains to all neighboring countries' capitals (Belgrade, Budapest, Chisinau, Kiev, Sofia), as well as to Vienna, Venice, Thessaloniki, Istanbul, Moscow and of course to main cities in all of Romania's 41 counties.
All international trains and most long distance internal trains arrive at Gara de Nord (Northern) station, located quite near of the city center, to which it is linked by subway and several buses, trolley, and tramway lines.
Some trains to and from the Black Sea Coast use either Gara de Est-Obor (Eastern) station, or Baneasa station,as well as the main Gara de Nord station; currently the route between Bucharest and Constanta,the main city in the black sea area is undergoing modernisation and you shoud expect long delays so if you can it's best to go by the Gara Obor station because it saves around an hour on your trip.
The other three smaller stations (Basarab, Progresul and Republica) are used exclusively for local and regional trains.
The timetables for domestic routes are available here:
Do not use any exchange services around the train station: they offer about 30-50 percent below the actual exchange rate--use an ATM instead or walk a few blocks to get a much better rate, then take the subway system, which is reasonably priced and has clearly marked maps and schedules. Watch out for the shady private taxi services and avoid taking taxis near the stations, no exceptions! It's not the case that a few of them may cheat you, but most are honest. You should know that near the stations all of them will try to cheat you and you will have to be both vigilant and lucky to avoid being ripped off. Always look to see if the cab driver starts the meter and alert him by saying "aparatul" (ah-pah-RA-tool) while pointing at the meter. There will be drivers offering rides - be extremely wary.