Heathrow to Cambridge

Cheapest looks like the Piccadilly line tube to Kings Cross and then the train. That seems to be 25 GBP without a student discount. The National Express coach would be easier - no change - but is 30 GBP for a random date in May. Fastest is probably Heathrow Express to Paddington, then the tube to Kings Cross and the train.

I agree with Kathy that Tube and train is the best option. The Tube fare from Heathrow to central London is £5.00. The fare from King's Cross to Cambridge on First Capital Connect trains is £20.00. The fare from Liverpool Street to Cambridge on National Express East Anglia trains is £14.70. Take the Piccadilly Line from Heathrow to Holborn. Transfer to the Central Line and take it to Liverpool Street.

Quickest and cheapest are uncomfortable bedfellows, even for students going to Cambridge. I nearly agree with the cheapest alternatives previously suggested. If these students are between 16 and 25 years old and they will be at Cambridge for a term or so they can purchase a "16-25 Railcard" which can take large lumps off train fares with some significant restrictions. If they open a student bank account with Natwest they can be given a 5 year version. On the other hand you also asked about quickest. I would expect that (barring a personal jet pack or helicopter) a private chauffeur waiting with a sign at the airport arrivals hall door is probably fastest, a taxi off the rank probably next. Neither of these is likely to be the cheapest.