thats ok, im relatively new to the area so spending this summer playing tourist anyway!
the town tours are great, there is a yellow bus and a red one, i have only been on the yellow but they both stop at a lot of the tourist attractions, the abbey, captain cooks museum, lifeboat museum, up on the west cliff where the go karts/outdoor pool/crazy golf etc are, also the whitby wizard museum there is meant to be great but i havent been yet. there are a couple of amusement arcades near the lifeboat museum. the yellow bus is £10 for a family day hop on and off type ticket so pretty good value.
elizabeth is a steam bus, the commentary is pretty boring in all honesty but my girls loved going on it and of course didnt listen to a word anyway!
the north york moors railway comes into the town a couple of times a day and there is info in the railway station about times for that
there is also a 60's bus that goes up to 'heartbeat' area (goathland) as a day trip, £15 for the family, only goes on certain days though, tue,wed,thur next week. leaves from the coop (really obvious between the train station and tourist infor) at 11.30am returning 14.45, im sure i picked that leaflet up in the train station too
coming out of town a bit there is pannet park which holds whitby museum and a brilliant play area, well worth a visit.
if you find (as it has been the last few days) the weather is bad then i would recomend doodle pots www.doodlepots.co.uk/ again my girls love it in there, i get asked every time we go into town. they are on skinner street, most people can direct you
if you want a cafe lunch in the week i would recomend going to the cafe in the colliseum, relatively cheap and lovely food with good portions, its off the tourist track but very good, its an old cinema i think, directly opposite boyes which should be pretty hard to miss. if you are down near the swing bridge the mulberry cafe is pretty good with some amazing cakes!
cant think of anything else but if you think of any questions shout, the weather has been wet but still relatively warm the last few days, how long it will last is another matter.
stock up with food before you come, the nearest tesco/asda/sainsburys are all in middlesbrough or scarborough, we have a large coop, large spar and a lidl. the butchers in town are good though for bbq food
hope you have fun