Bute's a cracking wee island to go around - nice quiet roads once you get out of Rothesay itself
You should look into doing the Three ferries ride
Fantastic day ride and some great views.
If the weathers good, I'm planning on doing
the six ferry ride. Port Bannatyne - Rhubordach - Colintraive (ferry) - Portavadie - Tarbet (ferry) - Claonaig - Lochranza (ferry) - round Arran - Lochranza - Claonaig (ferry) - Tarbet - Portavadie (ferry) - Colintraive - Rhubordach (ferry) - Port Bannatyne. Only about 200km, but due to the CalMac time table, it's a
ferry very tight schedule to do in a single day.