Winchester and Petersfield are both main stations, about an hour from Waterloo, with 2 trains an hour. However, they are 25 miles - 2 days walking - apart, with very little in the way of public transport in between.
In addition, these 2 stations are on different rail lines, which only meet up in Fareham (near Southampton), which makes for a more expensive ticket (2017: London-Fareham: £24.40 vs London-Petersfield: £18.20 - both Network Railcard prices), i.e. a ticket to Fareham is valid for both stations if you need to start your journey in one, and finish it in the other
Local Bus services [Jan 2017]
Stagecoach #67 (Mon-Sat only, every 3 hours) runs between the 2 train stations, but to the north of the SDW. Do check the times of the last bus. From West Meon they are about 18:45 (to Petersfield) or 16:50 (to Winchester).
Winchester (station) - Hinton Ampner (£4.50) - West Meon (£5.30 to Winchester, £3.50 to Petersfield) - East Meon - Petersfield Station.
There are rare #X17 (2 on Wednesday only) buses from Exton itself to Petersfield. You could use it to catch a bus to Exton, and then walk back.
These seems to change very frequently. Check! Traveline Southeast.