Behrens's monograph "The Goodwillie tower and the EHP sequence""The Goodwillie tower and the EHP sequence" reproduces some of the Toda calculations (out to the k~20 range as you cite) using a modern toolset, as named in the title. Depending on your tastes, the calculations may not be "simpler", but are organized according to two compelling relationships between stable and unstable homotopy.
Also, Neil Strickland has both written a treatment of Toda's methods and implemented Mathematica code to carry out calculations which you can find at http://neil-strickland.staff.shef.ac.uk/toda/