Hi again! Night life is not as diverse as in Moscow or to the west of Belarus but I am not sure about 'jazz' club even! The best approach to this side of acoustic ocean would be watching for events at Philarmonia or Concert Halls for classical or visiting jazz bands. Recently, I've seen several ads about American and european jazz artists/bands performing in the city and this summer there were open-air jazz evenings at City's Hall Square. There are three classical Saturdays' evening at City Hall Square coming this summer , you'd catch at least one of them. I can say that clubs are far from any modern serious electronic, IDM scene.
Mostly people leave the city for their dachas, country houses. Many spots within 50 kms from the city that are easy to drive to and you'd find yourself in a forest, near lake, river, small town with local market. 3 large parks in the city, many small green spaces, one large forested protected - natural Reserve area (Stiklevo) is right at the south-east edge of the city where you can bike/hike and even pick mushrooms, large water body with swimming in the city's north (Tsna) - all reachable in 40-50 minutes from city's centre. The city itself was re-designed in post-war period around central water artery (river Svisloch) with green zones and biking routes along it. It helps to funnel volumes of fresh air through city's core - in its area there are 200+ types of birds(nesting and visiting), beavers and adders among others live there too. I don't think you can do bbq in city's parks, but just outside it is all doable, although , watch out for neighbouring alco-consuming parties. Conflicts aren't likely but possible. 125 km from the city there is one of the biggest natural reserves in Europe, with local swamp/river/forest ecosystem and about 40 living brown bears(Berezinskij Reserve)- accommodation is possible too. Mushroom picking is a national sport and all neighbouring forests in the season usually become quite populated with those pickers.
Serbian is closer to Russian/Belarusskij than you may think!