Most people use the long-distance A/C bus services that run between Kolkata and Dhaka, with a change of bus in Benapole. Shyamoli and Greenline have offices here and on the Indian side of the border Haridaspur. For bookings see the Kolkata and Dhaka articles.
Alternatively, from Kolkata there are local trains from Sealdah station, taking around 3 hours to reach Bangaon. The border is 6 km beyond this station. Local buses may exist but good luck finding them. However, autorickshaws and rickshaws will take you from Bangaon station to the border.
In Benapole, the bus stop is about 2 km from the border crossing. From there buses head to Jessore Tk 30, 2hour and beyond.
If you have booked the bus from Kolkata to Dhaka,it will take 4-5 hours to reach Haridaspur on the Indian border side which is commonly known as Petrapole. The bus operators have their own rest houses, but their staff is not cordial and these toilets are dirty. Before going to Immigration please be sure that it is mentioned by Bangladesh High Commission that entry/exit is by road at Benapole. The Bangladesh visa is stamped and other entries are hand written, so be careful about any overwriting/cutting. If you find this when you collect your visa, please ask them to make an initial/signature and stamp over the overwritten part. Otherwise you are likely to be in trouble at the border. Customs people and immigration people at both the sides are cordial compared to their counterparts at the India-Pakistan border at Wagah. You may find that the Bangladeshi officials are better behaved than the Indians. The Indian immigration officials commonly demand bribes. However these problems may not occur if you are travelling by the direct buses. Customs usually do not present problems, except perhaps for Bangladeshi citizens returning home.
