Church Stretton is a small town and almost everything is walkable from the station. The High Street, the church, and the main cafes are all within five minutes on foot.
For the hills, you simply walk out of town. The National Trust car park at Carding Mill Valley is a 15-minute walk from the station, or a short drive if you're based with a car. The Long Mynd plateau itself is a car-free haven once you're up — the single road across it is narrow and discouraged for casual use.
Cycling is popular on the country lanes around Church Stretton and down the valley towards Little Stretton and Acton Scott. The hills themselves are serious mountain biking territory. There is no dedicated bike hire in town; bring your own or hire from Shrewsbury.
Taxis are available but in limited supply — book ahead, especially for early morning trains.