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Toda no doubt made some big strides when computing unstable homotopy groups $\pi_{n+k}(S^n)$ for $k < 20$ which his collaborators later improved upon.

The methods he used are documented in his book: "Composition methods in homotopy groups of spheres". However I find the book quite archaic and old-fashioned.

Is there a survey of Toda's work using modern language? Especially any modern insights that simplify his computations.

I find that unstable homotopy groups are neglected to some extent in modern work as there is a lot of focus on the stable case. And apart from Toda's work I cannot find any detailed surveys on these computations.

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Behrens's monograph "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/

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