Katowice, together with 40 nearby cities and municipalities, is part of The Metropolitan Association of Upper Silesia and the Dąbrowa Basin, also called Silesian Metropolis. The connection in the Metropoly is well developed and in some cases it's even possible to travel via train to a different part of the city, just like with trams or busses.
In order to travel inside of the Silesian region we recommend using Kolejne Śląskie - Silesian Railways.
The tickets can be found and bought here:
https://bilet.kolejeslaskie.com/bilet_relacja.jsp
Or via applications like:
SkyCash
KOLEO
The main office is situated in:
ul. Raciborska 58, 40-074 Katowice
The price of the tickets depends on the distance.
PKP
Trains that travel between the regions (i.e. Katowice - Warsaw) are mostly operated by PKP InterCity. However other providers can be found as well.
Their website and tickets can be found here:
https://www.intercity.pl/en/