The best museum in Glasgow is the Burrell Collection, but it is quite a way out of the town - a public bus stops by the entrance to the park it is situated in, then you walk about half a mile to the building itself.
Also in Glasgow, Kelvingrove Museum and Art Gallery is interesting if a bit manic, and The People's Palace has a fantastic ambience.
In Edinburgh, the National Museum of Scotland is very interesting but its top-floor restaurant is completely snobbish and up itself:
http://www.nms.ac.uk/