Museums/Art Galleries/bookshops often work pretty great. You're not stuck on the opposite side of the table from each other and you can skip the awkward small talk because there are literally hundreds of things to talk about.
It'll also let you get a good feel of what she's interested in and what your common interests are.
If it's going well - go for a coffee afterwards. Or a drink. Or a walk down by the Clyde. |