There is good reason to be interested in the culture and festivals in Bangladesh: despite its youth as a country, Bangladesh actually has the prestige of cobwebs all over its culture already. Bengali writers such as Rabindranath Tagore and Kazi Nasrul Islam have added to the richness of its literature over the years and strong interaction with the peoples of abutting countries, India included, have led to a genuinely intriguing local flavour.
One thing that is remarkable about Bangladesh is perhaps how curious the locals are about outsiders (or tourists). This is not a place where people surround the foreigner strolling through the markets in order to try to lead him or her to a shop for which they have a commission: it is far more likely for Bangladeshis to surround you just because they find you so unbelievably intriguing. Given that most of the tourism promotions in Bangladesh in the past few decades has been largely domestic, international tourists are not common in the country, which leads to most of the population-especially those in rural areas-still being terrifically impressed whenever someone clearly from out of town (like a Caucasian, for instance) shows up. Fortunately, the people are generally friendly, although they may be quite timid if you cannot speak their language. If you are going to the rural areas, expect them to follow you about like quiet and nice versions of paparazzi.
Bangladesh has a lot of festivals you may be interested in seeing. Hindu festivals such as Durgasatva and Buddha Purnima are observed, if you happen to be interested in them. This being a largely Muslim country, you can expect all the major Islamic holidays to be celebrated too, of course. Eid-e-Miladunnabi, Eid-ul-Azha, and Eid-ul-Fitr are all celebrated, as is Chaand Raat. Chaand Raat is actually a great time to see the Bangladeshi streets since they are sprinkled with lights then. Another good festival to see if you like colour in your festivities is the Shakrain Festival, which is held around the middle to end of winter in Bangladesh, around January: this is the time when the locals, specifically those in Dhaka, bring out hundreds of colourful kites and fly as well as fight them in order to celebrate the onset of spring.
Another festival to look out for is the Pohela Baishakh on the 14th of April: in the Bengali calendar as well as several other Asian ones, this is actually the first day of the year. This is perhaps the best of the festivals to attend in the country should you wish to avoid additional considerations of religious roots and understanding. What you get during Pohela Baishakh is basically a huge display of native Bengali culture, foods, dances, aesthetics, and fairs as well as markets. It is celebrated all over the country, like Independence Day, which precedes it and is held each March 26. This is rather a more subdued holiday, although one that still sees a lot of fairs and parades.
The culture and festivals in Bangladesh tend to mirror those of India, as may be expected from the countries’ histories. As such, anyone who has ever travelled to India may find Bangladesh almost familiar upon first setting foot in it. It is a largely Islamic-and-Hindu country with almost surprisingly open-hearted people who insist so much on showing hospitality despite their financial circumstances that you yourself might end up having to curb their attempts to show you the best time they can think of showing to a traveller.