The island is easily reached by car or train from Glasgow to Wemyss Bay (approximately 50 minutes), then Calmac ferry.
Calmac website: https://www.calmac.co.uk/article/7826/Bute-Wemyss-Bay---Rothesay (or click here).
The ferry journey from Wemyss Bay to Rothesay is approximately 35 minutes.
On arrival in Rothesay the flat is a two-minute drive or five-minute walk.
(If you're coming from the north, there is also a ferry from Colintraive to Rhubodach, at the northern end of the island.)
If coming by air to Glasgow airport, it’s a 15-minute bus ride (757) to Paisley Gilmore street railway station, from where you can catch a train to Wemyss Bay.
For bus details go to http://www.mcgillsbuses.co.uk/resources/site1/General/757%20TT%20April%202019.pdf (or click here)
(Car hire is also available at Glasgow airport)