You can go to Sylhet by bus, train or plane from Dhaka. AC buses of Green Line, Shyamali, Saudia, S Alam and Ena Paribahan travel from Fakirapul, Saidabad and Mohakhali bus stations in Dhaka city. AC bus fares are usually between 800 to 1200 taka. Also, non-AC buses of Shyamali, Hanif, Unique, Ena Paribahan from Dhaka to Sylhet will cost 400 to 470 taka per person. Get all the information about the bus from Dhaka to Sylhet, including the name of the bus, the amount of fare and the phone number of the counter, here: Dhaka to Sylhet Bus: Ticket Price & Contacts.
To reach Sylhet by train from Dhaka you can choose Parabat Express, Jayantika Express, Upaban Express or Kalani Express train from Kamalapur or Airport Railway Station. The train will take six and a half to seven hours. Take a look at this article to know the details of train journey from Dhaka to Sylhet: Dhaka to Sylhet Train Schedule.
If you want to go to Sylhet in the shortest time from Dhaka, you can go by plane. Biman Bangladesh, Regent Air, United Air, Novo Air and US Bangla Air depart from Shahjalal International Airport to Sylhet every day. Ticket prices from Dhaka to Sylhet vary from 3200 taka to 7200 taka.