(If you don't like links to other websites, simply ignore those that follow. They're only there to indicate my sources. All of the information you require is in this post).
There's a special bus service from Cheltenham Spa rail station for The Festival and for The Open meetings. At other times, take a Service D bus from the station forecourt. (The destination sign will show 'Bishop's Cleeve'). The buses run every 10 minutes and take 22 minutes to get to the racecourse.
This information is based upon the information on the racecourse's website:
http://www.cheltenham.co.uk/racing/getting_her e.html
and on the timetable here:
http://www.stagecoachbus.com/timetables/DApr07 .pdf
First Great Western operate from Paddington to Cheltenham, with most journeys requiring a change at Bristol Parkway. (There's also an infrequent direct service via Swindon, Stroud and Gloucester). Departures are roughly every half hour, with typical journey times between 2� and 2� hours, irrespective of the route taken.
Check your journey details here:
http://www.nationalrail.co.uk/
Chris