https://www.campinginthenewforest.com/campsites/ashurst/ is very close to Ashurst train station.
Co-Op a short walk north east where we discovered amazing spicy multi-cheese loaves ten years ago.
A bit further that way is Deerleap Lane which has the Owl & Otter Sanctuary (now called New Forest Wildlife Park).I’ve not ridden it often, but said lane isn’t bad to cycle on as regards traffic levels.
Not that far from Motor Museum.
Used to be a tour bus, no idea if still running.
Blissford Hill isn’t far away from there, never did get around to trying that monster before long covid!
Not ridden it, but I think Woodlands Rd almost opposite campsite is quite popular with cyclists.