Pretty much every regional train (not the white express trains) that runs from Cologne through Bonn stops at Bruhl. It takes about 15 minutes, and once you walk out of the station there it is: Augustusburg Palace. The main town is behind and to the right of the palace, if you want to do some further exploring. But the Palace and the gardens to the left and behind are the biggest draws in the town.
If you don't have a car, the easiest way of getting to Bruhl is by train. From Bonn or Koln (Cologne) trains are reaching Bruhl every 30 minutes. The trip is around 15 minutes and it leaves you just in front of the palace. Go out of the station and you will see signes for Phatasialand buss.
It is relatively easy to get to Bruhl from Cologne. The bus timetables are very easy to understand and the railway staff are very helpful if you are in doubt. One suggestion is just to give yourself plenty of time to study the times and you will be fine.