Ninewells Hospital Dundee
If you are referred on to Ninewells from the Community Hospital in St Andrews then you will need to change buses at St Andrews Bus Station.
Walk to stop (V) or (W) - there are six buses an hour from these stops to St Andrews Bus Station where you can catch a 99 to Dundee every 10 minutes.
In Dundee you can choose to either get off the 99 in Nethergate and cross the road to catch the first Xplore Dundee bus to Ninewells - services 5, 9 and 22 all go there and provide a frequent service.
Or if you prefer to sit down out of the weather to wait for your next bus, continue on to Dundee Bus Station and catch the next Stagecoach Bus to Ninewells - the 16 goes from the Bus Station to Ninewells and there are also 39 & 73 buses from outside the Bus Station in Seagate.
If you have to pay for your ticket the Stagecoach only Dayrider ticket is the cheapest option.
From Crail, travel to Ninewells is similar - catch any 95 to St Andrews Bus Station - change there to the next 99 to Dundee where there are frequent buses to Ninewells.
Fife Council have a leaflet on buses for Ninewells: https://www.travelfife.com/travel-fife/maps
Ninewells to Crail
You are recommended to catch any STAGECOACH bus from Ninewells towards Dundee as they all either go to the Bus Station or stop in Seagate opposite Dundee Bus Station. From Dundee Bus Station take the next 99 to St Andrews Bus Station where you will need to wait for the hourly 95 to Crail.