Now that it's been running and established a while, I visited the Wiltshire and Swindon History Centre's web site, and their "how to find us" page.
http://www.wshc.eu/visiting-the-centre/how-to-find-us.htmla) detailed instructions from five directions by car
b) Trains -
Chippenham railway station is operated by First Great Western (see First Great Western). It is on the main line from London Paddington to the West Country. Please contact train operators for details of services.
Which isn't quite accurate and ignores the TransWilts (they are probably right to ignore the TransWilts as a commute from Salisbury up to Chippenham would involve an arrival at 07:30 and a return at 19:00 - to an 09:30 to 17:30 facility - but it's practical on a Saturday)
c) Bus -
Bus services from Calne, Devizes, Swindon, Bath, and Trowbridge, amongst other places, are available to Chippenham.
The easiest option may be to take a bus to Chippenham railway station and then get a taxi or walk. If you choose to walk, it will take about 10 minutes.
No mention there of Salisbury, Melksham, Warminster (all of which the centre covers, and are larger than Calne or Devizes).
[i]Of course it is easy to be critical of which places / options are selected ... it's not clever on the departure sheets at Chippenham, which include direct services to Melksham and Westbury, but not Trowbridge, not Warminster and not Salisbury these days.