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 a LINK to 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 a LINK to the timetable for that route: