We cycled this route with our two teenage children (13 and 15 years) so did it over 6 days cycling with our longest days being 38 miles. We took the bikes on the trains (Northern trains to Morecambe and from Bridlington let us take all four bikes on their trains with no problem (we had to travel in pairs on the Cross Country trains and reserved the bike spaces - you can only book two bike spaces per train and they only have space for three bikes).
It is the most wonderful route - the Yorkshire Dales are very hilly, constantly going from easiest to hardest gear as we went up and down, but it is stunningly beautiful and I only had to push up a few of the steepest hills (which I was pleased about given we were carrying all our things in panniers). Each place we booked depended on them having secure bike storage. We stayed the first night at the Balmoral Guesthouse on the seafront in Morecambe which had beautiful sea views and then cycled to Settle (stayed at the Craven Arms in Giggleswick), then we had the hilliest day to Padley Bridge (where we stayed at the Crown Inn, again recommend - fantastic family room), then a short day to Boroughbridge with stops at Brimlow Rocks, Fountains Abbey and Ripon. We stayed at the Black Bull in Boroughbridge (not my favourite stay but great beer). Then another short and easy flat day to York (we wanted to spend the day in York and stayed at 34 B and B which was almost on the route and did early check-in (11:30-1:30)). Then a longer day through the Wolds to Hutton Cranswick (stayed at the White Horse) and our final day to Bridlington.
The route was so beautiful and wound through beautiful villages and followed really quiet roads. It amazed me that we could cross the country with so little traffic, cycling right into the centre of towns and out, through Fountains Abbey deer park and right past York Minster. We loved this trip, especially the Dales; it had so many contrasts and was well signed all the way.