Hi:
I do this trip all the time and it's actually one of the easiest from Heathrow.
Take the RailAir link direct from Heathrow to Woking Station.This is a half-hourly coach service from all terminals and takes about 50 minutes to get to Woking.You can buy tickets for this at each terminal.
At Woking Station you can pick up a direct and frequent South-West train to Yeovil Junction.You buy tickets at the station.
www.southwesttrains.co.uk/SWTrains

There is no need to go to London,Reading or Basingstoke.

If you have time I recommend a visit to Salisbury which is a short distance on the same train route from Yeovil.It's a nice cathedral city not far from Stonehenge.
If you have time stop into the Haunch of Venison for a pint of Summer Lightning.They've been serving beer here since the 13th Century.
www.haunchofvenison.uk.com/

Good luck.