For a one time only job you may be best contracting the work. Be sure to ask him about the accuracy. There are also large companies that do this kind of contract work. To do it yourself you will need either a scanner big enough to accomodate your map or a digitizing tablet. With a scanner you will need raster to vector conversion software and then a vector editing software to clean up the resulting drawing file.
Some colored maps can be problematic.
With a digitizer, you need software to control the digitizer. I well understand your problem. I have just completed digitising a few maps, but fortunately I did not have to digitise so many lines. The only options that I see are, either digitising or scanning. But scanning is not so simple either, if you do not have a very clean map with no extraneous lines and other features.