I normally sit down and make a drawing of the part. This may take the form of a rough sketch with dimensions with pencil & paper for a simple part to a very detailed CAD drawing for something more complex.
The measuring is normally done with digital calipers, but I also have a small height gauge at my desk which simplifies some dimensions. Normally I will find the hole centers by measuring both hole edge to hole edge, then do the math to find the centers. The get hole edge to part edge in both the X and Y axis.
For maximum accuracy I use gauge pins in the holes and measure over those, then use a surface plate and height gauge or indicators for deriving the hole to edge distances. Measurements may also require the use of radius gauges and angle gauges.
How you approach the job depends on how accurate you need to be, using calipers will normally get you close enough for most work.