What's the best way build surface around vertices? I find it very time consuming to select vertices one by one and use bridge or fill tool to create faces.

Is there a better way to fill things similar to the one in the screenshot? Such as creating a plane then cut it with the pattern to be filled around?
Try the F2 addon. It's already built into Blender, but isn't enabled by default. When it's active, pressing the 'f' key with a vertex selected extrudes an entire face from it instantly and allows you to continue adjusting its position afterwards. Really speeds up modeling and retopology efforts.