The CalMac ferry from Ardrossan on the Ayrshire coast serves Brodick, with crossings taking 55 minutes. Ferries run several times daily year-round; more frequent in summer. Book ahead in July and August, particularly for vehicle spaces — the ferry fills up on summer weekends and bank holidays. Ardrossan is about 45 minutes by train from Glasgow Central.
The alternative crossing is from Claonaig on Kintyre to Lochranza in the north of Arran — a 30-minute ferry, running summer only, that avoids Glasgow entirely if you're coming from the west or north.
There is no bridge to Arran. That is part of the point.