What is the best way to export textures with objects?

"Best" will always be subjective. In my experience, it's better to not export them at all. This is because for all but the simplest textures, you'll want to maintain the image as a high resolution layered image (PSD or XCF, say), which Blender can't understand anyway. I generally have a PSD files with layers for the UV layout, AO (usually I just do that by hand, but Blender can bake it), a bump map (again usually by hand) that Unity can convert to a normalmap, and logical layers for the diffuse texture itself. I save out various combinations of layers to PNG files (sometimes the PSD is saved as the diffuse), which I use in Unity and for fine-tuning within Blender.

The exception to this that I have found is if you want to paint inside Blender, though I don't find that very comfortable ... YMMV.

The simplest would be to export an FBX into your project directory.

Make sure your textures live in a sub directory called textures - before you export them, then Unity will create materials for them at import.

Blender files are also natively supported, in either case there is documentation in Unity Manual Here:

http://unity3d.com/support/documentation/Manual/HOWTO-ImportObjectBlender.html