Don't go alone, tell someone where you're going and time you expect to make contact by to say you're out safely (allow time to track down mobile phone signal), and make sure you have spare lights/batteries with you. and spare clothing in the car. It's a popular first cave trip, but most people on their first cave go with a more experienced caver.
Use the 1:25000 OS map. Park in layby a bit higher up than Hill Inn. Obvious footpath next to small Water Board building, with an information board just inside gate. The path is one of the main route up Ingleborough so is quite obvious. After second wall there is a cross-roads of paths - I think there's even a signpost, though I've no idea what it says. Take the left, a clear path running between banks. After about 300yds you come to a huge shakehole with trees. Go over the wall at the obvious stile and follow the path to the bottom.
At the bottom you'll see a bit of scaffolding and a dig - ignore this. Great Douk Cave is where the waterfall comes out at the head of the shakehole. Just clamber up the waterfall and explore.
Main passage is obvious, just behind the entrance to the left is a less obvious branching passage which is interesting to explore. Main passage is fun, eventually lowering to a crawl in water - quite a way along here is a hole in the roof which pops you into another series of small passages - bear left whenever possible and you emerge from Middle Washfold Cave. Or just turn round and go back the way you came in.
Search on 'Great Douk' on this forum - there was a lot of discussion on route finding when another person used it a their first cave a few months ago. And if you enjoy it, the Yorkshire bit of the forum has info on other suitable caves.