I'm not a fan of partial classes - I try to develop my classes such that each class has a very clear, single issue for which it's responsible. To that end, I don't believe that something with a clear responsibility should be split across multiple files. That's why I don't like partial classes.
With that said, I'm on the fence about regions. For the most part, I don't use them; however, I work with code every day that includes regions - some people go really heavy on them (folding up private methods into a region and then each method folded into its own region), and some people go light on them (folding up enums, folding up attributes, etc). My general rule of thumb, as of now, is that I only put code in regions if (a) the data is likely to remain static or will not be touched very often (like enums), or (b) if there are methods that are implemented out of necessity because of subclassing or abstract method implementation, but, again, won't be touched very often.