tyleej, Zürich, like all other Swiss cities, is super easy to get around. It has a dense and frequent network of trams which will take around the city centre and the rest of the city.
If you are arriving by train, you will arrive at the main station, "Zürich HB", which is on the edge of the centre. The centre is small enough to walk around, relieved by tram rides when you get too tired.
I would not bother with the coach trip, the old town is more fun on foot. Most is pedestrian only (+ trams), so the coach couldn't go there. The other places you list on the coach tour are all accessible by public transport.
Basically, everything is accessible by public transport and on foot. The Swiss do not like cars in their city centres.
How and by what method of transport (air, rail) are you arriving in Zürich? If you are flying in, you could go direct to Luzern. It is only 70 minutes by train from Zürich airport.
Is it easy to do by train and find the boat company to take us to Pilatus?
Yes, super easy to get anywhere in Switzerland by train.
In Luzern, the boats leave from the centre of the town, in front of the railway station.
For info on the Pilatus trip, see here: https://www.pilatus.ch/en/discover/golden-round-trip/