I realise that lots of people give generic advice of things to do in the Yorkshire area without ever being very specific so here is my attempt to combat that. If you have any suggestions or ideas please feel free to add, as I will do when I think of them.
The Beach (Including the amusements, one can make sand angels, sand castles or on rocky beach areas go rock pooling)
The National Railway Museum (This is free and actually quite enjoyable, they have events on during school holidays and other special events like Christmas if you wanted to utilise your day out)
The deep (I know what you’re thinking it is just a bunch of fish, but once you go once you can convert your ticket for the same price into one that lasts the entire year so multiple trips can be made here after the inital outlay. This aquarium is well worth a visit and you can save 10% on tickets if you book online)
Check out the museums quarter, ferens art gallery and the maritime museum all free entry and all worth a look at once.
The Reel cinema (Reel Monday all tickets are £3.90 2D, although they do not follow Orange Wednesday this is a good option for those busy on a Wednesday wanting another cheap day at the beginning of the week)
Xscape – Can be expensive but it depends on what activity you pick, and can result in a fabulous day out.
Monks cross is always a good one for shopping.
The national railway museum which is conveniently located close to the new railway is free entry and has extra special events on during Easter holidays, summer holidays and Christmas for the children.
The Wolds Way – Starting in Hessle (near Hull) and finishing in Filey (near Scarborough) is a easy walk for all levels although a map is recommended and costs nothing more than food for the journey and perhaps a tent for the evening or a B and B. It can be done in under 24 hours but 4-5 days is my recommendation in order to take in everything.