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 Haridaspur. Local buses may exist but good luck finding them.
In Benapole, the bus stop is about 2km from the border crossing. From there buses head to Jessore (Tk 30, 1 hour) and beyond.
You're here to do one thing: walk 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.
Get your passport stamped - The Indian side is fairly normal as far as land border crossing 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. 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 almost 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.
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 taken about 4 hours to produce. Sometimes you might need to insist that the office really does provide these certificates and demand that you need one.
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.
If you must sleep at the border it's best to do so on the Bangladeshi side.