Getting to and from Arran is simple enough. If you are thinking about a longer tour in the summer it might be worth going one way on the summer ferry between Lochranza and Claonaig. On the other hand the ferry to Brodick from Ardrossam operates all year and there is a train between Ardrossan and Glasgow. There are only 4 or 5 train/ferry combinations a day each way and the last one is not very late so it is imperative to check current timetables. These appear at the end of the bus timetable shown below.
On the island there are circulars for the whole coastal journey [about 55 miles]. There is also what is called the String road from Brodick to Blackwaterfoot across the interior of the island. £3 Rovers [Oct 02] allow for unlimited use of the services on these lines. You can find the timetables for these and other bus services on the island as well as connections by ferry and rail to Glasgow if you look at http://www.stagecoachbus.com/western/db_search.html and put 'Brodick' in the Search Box. The search result will allow you to download the timetables.