I grew up in Hull and Beverley, so know the area well. As above, the Docks and Land of Green Ginger are nice, as is the Marina. The Deep is great, and also try the Town Docks and William Wilberforce Museums. Princes Quay Shopping Centre is worth a mooch, especially the indie shops on the top floor. Beverley has a lovely Minster and some nice shops. If its sunny, and you have a car, the East Coast has some nice beaches, especially Spurn Point.
I dont know if Spiders is still there…if you go, wear your crappiest clothes, its a dive, but great..and you have to try the PanGalacticGargleBlaster and the Wiggly Worm, their own creations which were (are?) dirt cheap and guaranteed to make you fall over. Wipe your feet on the way out though. The Beverley Road pub crawl is another fave.
There isnt much in Cottingham, its just a (very) big village – local rumour is that its the biggest village in Europe, whether its true is anyones guess. Its Student Hell otherwise.
Dalby Forest is only an hours drive away……