Hi, welcome to Romania ^^
If you have the time, try to go around the big parks, they are plenty and really nice to walk through, Cismigiu, Herastrau, and Carol I parks are the best.
You could see the Parliament Palace if you wish to see a massive building commissioned by the last communist ruler.
The village museum is something, in my opinion, any tourist should visit.
The Cărturești Carusel bookstore is a must-visit if you like books, but it's not only about books, you can even just enjoy a coffee while looking around.
And lastly, although I feel like I'm missing something, you have to try some traditional foods, my top recommendations would be "mici", which are caseless sausages that are absolutely delicious, and "sarmale", which are cabbage rolls. And as a dessert, try "papanasi", which are fried cottage cheese doughnuts topped with sour cream (that's not sour) and jam, usually blueberry jam.
I hope you will enjoy the city :D