Khan Jahan Ali was a famous 15th century Sufi that settled here and established the town, originally known as Khalifatabad. His tomb and mosques constructed in his honor are the main reasons to visit. While none of the sites are likely to blow your socks off it's still a relatively peaceful village to walk around.
The local area code is 401.
A cycle-rickshaw from the bus stand to the Shait Gumbad Mosque should run around Tk 10.
Unless you're in a hurry, you may want to just walk to the sights from the bus stand. Head west along the main road, a fairly nice walk save for the speedy honking buses. The first sight you'll come to is the dargah; start asking around for it after about 5km or so. Shait Gumbad is another 2km or so west along the main road from the dargah turnoff.
There are several other mosques along or just off the main road along the 5km stretch west of the bus stand. Most are single domed and peaceful but not all that interesting unless you have a specific interest in mosques of this style and period.
There isn't much to buy here aside from some prayer beads and other little trinkets near the dargah.
The only options are some basic dhabas near the bus stand and near the dargah that serve the standards. - you may want to do a quick health inspection of the food before deciding on one.
There's a couple of lackluster budget hotels just north of the bus stand near the river. There isn't much reason to stay the night here as the sights can be visited within an hour or two and Khulna and Mongla are both so close. If you must stay, pick one and it'll likely run around Tk 50-70 for a single or Tk 120 for a double.
Khodla Math - this Hindu temple is not so interesting but may be worth a sidetrip if you want to kill some time. It was built by a Brahman in the 17th century, but is not standing the test of time very well. It's near a village named Ayodhya, about 10km or so north of Bagerhat. A rickshaw will drop you in Jatrapur, from where it's about a 3km ask-directions-as-you-go walk.