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 2 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 Indian officials demand bribes. On the other hand the Bangladeshi immigration officials are better behaved than the Indian officials. However these problems may not occur if you are traveling by the direct buses. Customs usually do not present problems, except perhaps for Bangladeshi citizens returning home.
You're here to do one thing: walk / travel across the border. On the Bangladeshi side, you can walk the 2km to the bus stop or take a rickshaw for Tk 5.
On the Indian side the bus stop is near the immigration office. As you step out of the immigration office to the official moneychangers stalls, there may be people sitting around who can arrange a taxi to Kolkata. I paid Rs 900 in Jan 2008, but usually it will be more.
Get your passport stamped - The Indian side is fairly normal as far as land border crossings are concerned. Your passport will change hands several times, and should finally result in a stamp, which doesn't take too long. The Bangladeshi side, however, is another story. Staff seem to have been hired an hour before you arrive - try not to laugh when they ask which visa in your passport is the Bangladesh one. On either side you may be asked for baksheesh by the passport stamper or a boy who grabs your passport and shuttles it between the various officials or the many men who offer to fill your forms in for you. This is not normal for an Indian border crossing, and is entirely avoidable - shuttle your passport yourself, and if still asked, no works. If you're on one of the A/C direct buses then the bus company collects all passenger's passports before the border and facilitates the stamping.
Recently the process has improved considerably. The Bangladesh side is simpler than the Indian side where you wind you way through a complex building to get the various stamps. On the Bangladesh side, you will enter one office, be asked to sit while you passport is examined, and within short order, stamped.
While leaving Bangladesh by road, you have to pay a departure tax of 300 Taka. If you are travelling by one of the direct buses, the bus company will usually collect the amount from you and pay it for you. If you are travelling independently, pay it at the little branch of Sonali Bank next to the immigration office. It seems to be open as long as the border crossing is open.
Note: if you arrived by air to Bangladesh, make sure that you have a "change of port" certificate which allows you to leave by land, otherwise you will find yourself being turned back at the border. Change of port certificates are available from the visa office in Dhaka and take about 4 hours to produce. Sometimes you might need to insist here that the office really does provide these certificates and demand that you need one.
comment by Felix: Another border post is Darshana/Gede which is quite convinient by train. I crossed the border coming from Bangladesh 2 times. Once on the "Maitre" direct train. I wasted half day to get the "change of port" but at the border nobody asked about it. In September 2015 I travelled again taking the train towards Jessore, exit at Darshana. Take a riksha, pass by at the bank to pay the travel tax and continue to the Bangladeshi border post. I had a multiple Bangladeshi and Indian visa and arrived by air in Dhaka. Again nobody asked about the "change of port". I saw Indian visa which is limited to certain border posts. Same might be possible for Bangladeshi visa but if the visa has no limitations there should be no problem and no need for "change of port".
Money changers are much more plentiful on the Indian side. They usually offer decent rates, but may try to make up for it in other ways - count the cash you give and receive carefully, and do the math yourself on one of their calculators.
There are a handful of food stalls and dhabas on both sides of the border serving local Bengali food, and a couple hotels on the Bangladesh side have restaurants.
Don't drink anything except sealed packed water bottels.
If you must sleep at the border it's best to do so on the Bangladeshi side.