The Great Western route from Paddington is the one that most people use to Cornwall as it is the fastest. However, another route exists from London (Waterloo) to Exeter and it is this route - operated by South-West Trains - that goes through Whitchurch (Hampshire). As this is a village, it is only the stopping trains that call (hourly) & not the trains that continue to Exeter.
Go to www.nationalrail.co.uk & put in for WAT to WCH. You will probably find the single fare is £27.80 - which is expensive for the distance covered. (Returns are often virtually the same price). It does not look like cheap advance purchase fares are available to reduce this but you may like to consider buying a railcard to reduce the fares but beware that these might not be valid before 9.30am. http://www.railcard.co.uk
Anyway, assuming that you have seen Whitchurch & armed with the train times going west, you will need to go to Exeter (St.David's) in order to connect with trains going into Cornwall. You would probably be advised to change at Andover bout I would consider going from Whitchurch to Salisbury & once again leaving the train to visit the medieval cathedral - which is about a 15 minute walk from the station. Salisbury also has shuttle buses to Stonehenge so you might like to make this a 1 night stopover before heading off to the south-west the following day. Doing Whitchurch & Salisbury & reaching Cornwall in 1 day is pushing it.
If returning to London on the Great Western route, you would be well advised to advance purchase tickets. You may also like to consider going from Cornwall to Bath for a night or 2 before returning to London.
http://www.southwesttrains.co.uk
//www.firstgreatwestern.co.uk