Getting to the centre of Bratislava
The best option is to buy a public transport ticket and then use Bus #61. You go to the end stop which is at the railway staion ("Hlavna stanica," which means "main station"). It takes 25 mins, the bus goes every 20 mins.
Getting to Brno
Use the web schedule to plan for a suitable train. The ticket to Brno should be about EUR 7.50, and you should get this price when buing the ticket at the counter at the station. It should be similar for the oposite direction.
Group discounts for 6+ groups.
There are buses by StudentAgency, but they are slower and more expensive than train (2 hrs, CZK 220, i.e EUR 9.20, no discounts for <=26 nor ISIC). Reportedly, there is a long term reconstruction on the way from Brno to Slovakia, which adds extra 30 mins.