dąb
The old part is sh!t, the new apartments close to the shopping centre are nice though.
srodmescie
There is no place in Katowice called Śródmieście, however, there is an area of Tychy called that.
It's about a 30 minute drive from Katowice, in traffic it would be more like 45 minutes.
Personally, I can recommend the south city centre, also Brynow, Osiedle Tysiąclecia is good too, huge park and shopping centre close by, serviced by many buses and trams or it's also close enough to cycle to and from.
Also Piotrowice or Ligota are ok, but you'ld be better off having a car living out in those areas, buses service them, but not so many as Tysiąclecia.
There's a basic rule to Katowice, try and stay south of the train station and you should be fine.
Avoid Załęże, Szczopenice, Nikiszowiec, 1-maja, Zawodzie, they are all scumholes.