OK well here are a few thoughts (lived in Brussels for 3 years).
Near Grand Place, would suggest St Géry market hall area for a wander, bars etc (over road from Bourse). Also St Catherine which is nearby. Some of the streets further away towards the canal have some good bars etc in (less touristy).
Then there is essentially up the hill above that area so up to the Sablon, Royal Place, Palais de Justice etc which you could walk around as they are nice areas/buildings. Little side streets there also have some good bars/food stops in. This is essentially above Centrale/Centraal and the main railway line through Brussels. The musical instrument museum has a rooftop bar/café on Coudenberg. As it is up on the hill even from the street you can look over the lower town where Grand Place is.
If heading further East in the upper town (over inner ring road) then Ixelles (e.g. Chatelain, Flagey) and St Giles have nice areas too with squares, bars etc. Also as discussed elsewhere on the forums the older trams on some of these routes e.g. 81 back to Midi.
If you like a beer then of course the city has plenty to offer. A few quick thoughts there would be find a bar serving the local brasserie de la Senne beers
brasseriedelasenne.be. Very refreshing classic unfiltered Brussels style beer. For a range in a nice setting then the
www.moederlambic.com bars are good (one in St Giles, other on Place Fontainas just along road from Midi to Bourse).
Transport wise you can get to most of these parts on Metro/pre-metro. Also tram e.g. St Giles and more tucked away bits or if not wanting to walk then bus. However, I reckon you could choose a suitable metro/tram station/stop and then walk.
Hope that helps a bit and have fun!