There are quite a few clubs, recreation spots and hotels in Dhaka that have tennis courts, and most of them are free to members and those staying there, but outsiders can rent the courts for games. As for where you can learn to play tennis, there are some places in Dhaka for that too, and all of them have proper training programmes, professional coaches and trainers.
The best places to learn to play tennis would be Bangladesh Tennis Federation, Ramna Tennis Complex, Gulshan Youth Club, Dhaka Club, Nordic Club Dhaka and Uttara Club Ltd. Bangladesh Tennis Federation is the national institution and it charges 500 Tk per month for training. After you become a proper player, you become eligible for regional tournaments which can lead to national and international tournaments, but let's put those far-fetched fantasies aside for now and learn the game first.
You don't actually need much to start, except enough cash to pay for a membership and a pair of shoes. You don't necessarily need to buy a tennis racquet for the first day because you really wouldn't know what kind of racquet would be good for you. And at a store there would be so many types of racquets you wouldn't know which one to buy. And they're damn expensive too, so buying one that turns out to be the wrong one for you, would be a huge waste of money. The place where you choose to learn will probably provide you with a racquet for the first day, and then tell you what kind to buy with what specifications depending on your strength and ability.
Tennis shoes improve your speed, footing, bounce, grip on the court and they help you perform better, all while keeping your feet cosy. But as a beginner, I suggest you forget about good shoes and focus on learning the game, unless you can afford them without a problem because good tennis shoes come at a hefty price. Otherwise, play in average tennis shoes or sneakers; they work just fine. Besides, you should save the money you would use to buy tennis shoes for later, when you're actually experienced enough to know which shoes would benefit you the most.
As for any other sort of gear or equipment, you don't need anything at all. But if you want, you can get a wristband to reduce the chances of spraining your wrist and for comfort too.
If you love playing any kind of sport, you won't be able to dislike tennis. Learning tennis improves your basic skills at other sports as well. Don't bring up how Bangladesh is doing in international tennis and that there's no point in playing. I'm not telling you to play professionally; I'm just asking you to give it a try. Because after all, tennis is one of the most fun sports that exist; you should go find out for yourself.