There are two ways to get to Bute. Assuming that you are not coming by private jet or yacht, of course! *
From Glasgow you can either head straight down the M8 and head for Wemyss Bay and catch the ferry to Rothesay (takes about 35 mins). Here’s the timetable for that route.
The other way takes you the scenic route through Argyll. You will arrive at Colintraive where you can catch a small ferry for a very brief crossing (3 minutes!) to Rhubodach. From there just follow the road through Port Bannatyne and down to Rothesay. Here’s the timetable for that route: