Outside of Edinburgh there is Rosslyn Chapel which is one of the places suggested as being the final resting place of the fabled Holy Grail.
In Glasgow, there are numerous art galleries and museums the best of which are located in the west of the city. The Burrell Collection in the south is also worth visiting. For seeing pure crap, stolen car radios, and pirated software for sale then you can visit the world famous Barras market which is open on a Saturday and Sunday. Ingliston market, outside of Edinburgh, is inferior to this market.
If you fancy a day trip outwith the Glasgow and Edinburgh area then Ayrshire contains the poetic history of the Scottish bard, Robert Burns - his house, his friend's house, the kirk where Tam O'Shanter saw the witches dance a'widdershins.
If you are within the Glasgow and Edinburgh area then you'll probably find it best to spend a day or two looking around these places (Edinburgh is much smaller than Glasgow but has loads of little side streets that lead off and, at night, they do ghost tours, etc. while Glasgow is its complete opposite: modern, streets are open and of the gridlock layout, and has a great selection of night clubs, bars, and restaurants. You should, however, try and get as many day trips in as possible and visit the tourist sites: New Lanark as has been said, Arran ("Scotland in miniature"), Culloden, the Trossachs, etc.