Now that Mindstorms is on its third generation, the cheapest way would be to buy older sets with the oldest “intelligent brick,” the RCX.
There is a league for Mindstorms robotics called FLL or the FIRST LEGO League. If kids want to get into that I suggest they find a team.
http://www.firstlegoleague.org/parents/childonteam
Participants should expect to pay at least $100 – $150 to participate on a team of ten students because the team buys basic parts, the competition kit, (which is new each year,) a registration fee and a competition fee.