I wouldn’t cycle along Princes street myself, let along with kids, apart from the bus, tram and taxi interfaces, it is hoaching with tourists at the moment, even worse than usual (that will be the weak pound) who dart out randonly all over the road.
The trams do have a bit that says you can put a bike beside the doors, but I have never seen anybody do it, the area is quite small.
If you come in at Waverly you can board another train to Haymarket, they are considered the same destination. Get out there, and use the cycle path, which does eventually get you on the canal path, although further along from the basin. I am sure there will be a Spokes map online if you search.