Airport information
See the BTS airport webpage.
 Getting to the centre of Bratislava
The best option is probably to go by public transport: bus #61 starts at the airport and terminates at the main railway staion ("Hlavna stanica"). The journey takes 25 mins, the bus goes every 20 mins (details).
You need to obtain a 30-min ticket before entering the bus at the ticket-vending machines near the bus stop; then you have to stamp it immediately after entering the bus.
Tickets are paid in Euro coins. This may be the first occasion when you need Euros, which makes it a bit more tricky to obtain the tickets. If you have small notes (5 to 20 euros) you can ask someone at the airport for the micro-exchange. For 50 euro notes, you will probably have to buy something at the airport to exchange it.
 Getting to Brno
 By train
Use the web schedule to plan for a suitable train.
At Bratislava, "Hlavna stanica," both euros and credit cards are accepted (though perhaps not at every counter; follow the labels or ask).
Return ticket is EUR 9.9 (valid for 1 month), one-way is EUR 7. (Exceptions are the SuperCity trains and the EuroNight sleep trains.)
There are discounted group tickets: for return ticket, they are available for groups of 2+ people and the discount seems to be (n-1) * EUR 1.30 (obviously, the group has to stay together also on the way back). For one-way tickets, discounts are available for groups of 6+ people.
[SuperCity Pendolino] is a nice modern train; local people feel quite excited when using it (but if you are coming from France, you have no chance to get impressed...).
There is only one SuperCity connection currently: at 5.42 from Bratislava and back from Brno at 20.51. With return ticket, you have to buy a seat reservation for EUR 3 for each direction for which you are going to use the SuperCity train. For one way, you have to buy a ticket for EUR 10.40 instead of the normal one.
 By bus
There are buses by Student Agency, but they are slower and more expensive than train (2 hrs, CZK 220, i.e EUR 9, no discounts for <=26 of age nor ISIC). Reportedly, there is a long term reconstruction on the way from Brno to Slovakia, which adds extra 30 mins (otherwise it would only take around 1 1/2 hrs, I guess).