2nd go - the site froze last time and put me onto technology - which I couldn't escape. There is so much in Yorkshire (it's big), but it sounds as if you want to be quite local to Whitby. Don't know how much time you have.
Pickering is always worth a visit, nice little place but don't overlook the church(many do) which has some of the earliest frescoes ever recorded in England. There is the starting station for the N. Yorks railway and bookshops and pubs and coffee houses and... and... On the way there are Grosmont and Goathland, of 'Heartbeat' fame.
The 'Lyke Wake' walk crosses the moor, but I don't know of stopping places. Below the Hole of Horcum there used to be a good pub, but it was closed 2 years ago. Rosedale is a gem of a drive.
If you are reasonable active and interested in heritage etc. I would suggest a day out driving to:-
1 Scarborough headland and visit the church with the grave of Anne Bronte and good views, castle also interesting (but probably no time to go round).
2 Flamborough Head, impressive chalk cliffs and bird sanctuary for a walk, followed by..
3 Lunch at a gastro-pub called something like 'Seabirds' on main street in Flamborough village - good helpings, good beer, good food - was there early Oct..
4 Drift down to Burton Agnes (A614 after Bridlington, heading back to Driffield) and visit the stately home there. Excellent art collection, very interesting house, lovely gardens, good tea-rooms (smashing, friendly local guides).
5 Head back cross-country through some of the renowned Wolds villages (just put an offer in on a house in one of them) - Kilham,Langtoft, Sledmere (Hockney paints this a lot) up to Malton.
6 Malton - lovely little market town, worth a stop to see where Dickens stayed and got the idea for 'A Christmas Carol', church in market square is lovely, worth a walk round, lots of individual shops, pubs for meals etc.. Depending on timing you are now about 10 mins. from Pickering and driving back across the moors.
I could do lots of other itineraries - if you like wine you might like to look up 'Ryedale Vineyards' (the most northerly vineyard in England) just south of Malton and see if they have a wine tour. I can personally vouch for the wines and they have been winning medals and awards galore this year.
Hope this helps.