Finstock is on the Cotswold Line from Hereford to Oxford. There are two trains per day from the Finstock Halt (to Oxford at around 7.50am, returning from Oxford around 5.50pm). Charlbury station has a more frequent service throughout the day (though beware the mid-afternoon gap!). Finstock and Charlbury are around an hour and half from London Paddington / Worcester. Info: www.thetrainline.com
The X9 Witney - Chipping Norton bus service stops in Finstock - view the current timetable here. In Charlbury you can board the S3 to Oxford via Woodstock (or take a short walk down to Charlbury railway station for onward connections) - more info here. There is also a peak time railbus connecting Finstock commuters with Charlbury station.
Reach Finstock from the A40 / A4095 via Eynsham or A44 / B4022 via Charlbury.
There are many scenic routes in and out of Finstock by bike. Sustrans route 442 (Worcester to Hanborough) passes through the village and is clearly signposted (though beware the kissing gate you need to pass your bike through/over as you enter Cornbury Park en route to Charlbury).
Finstock is popular with ramblers and is on many scenic walking routes. The Wychwood Way, a 37 mile circular trail, passes close by the village.