A cx/mtb (with skinnyish tyres) would be ideal to follow the Whitby – Scarborough cycle path and the network of coastal roads and (whisper it) coastal footpath in that neck of the woods. This might be a nice way to spend time on the bike with someone who doesn’t fancy big road mileage. South of Whitby, Robin Hoods Bay and Ravenscar are worth a visit. North of Whitby (south of Staithes) there’s Runswick Bay and Sandsend to see. All good for coffee, cakes, ice-cream and what-have-you.
In terms of road riding along the coast the A171 can be very busy, difficult to avoid and, for long stretches, keeps you away from the coastal scenery. You might be better heading south from Staithes into the North Yorks Moors but here big hills may be off-putting.
Take a change of tyres for the cx (skinny slicks) and longer (solo?) road jaunts become a more comfortable proposition.