In some ways, the "best" approach to calculating the homotopy groups of spheres is to identify patterns in the homotopy groups of spheres, rather than trying to make a complete calculation of all homotopy groups up to a certain dimension. This "best" approach, led by Ravenel and collaborators, is to try to determine which families of elements in $ Ext = E_2 $ survive to $ E_\infty $ and therefore detect non-zero elements in homotopy. Often these families of elements in homotopy groups have an infinite number of elements, but the recent work of Hill-Hopkins-Ravenel on the Kervaire invariant is a significant example of a family of elements shown to be infinite in number in Ext but finite in number in homotopy (i.e., there are infinitely many non-zero differentials in the classical Adams spectral sequence on this family).

A complete calculation of the homotopy groups of spheres at any prime is difficult for many reasons. Since Ext calculations (using any suitable generalized homology theory) tend to be very large, neither humans nor machines can make a complete calculation up to a large dimension in a short amount of time. Also, calculating differentials often requires topological information not present in a single algebraic Ext object. It is common to compare an Ext calculation for one generalized homology theory (e.g., mod p homology to obtain the classical Adams spectral sequence) to an Ext calculation for another generalized homology theory (e.g., Brown-Peterson theory to obtain the Adams-Novikov spectral sequence) and see if any differentials are forced by comparing the two spectral sequences, but results using this approach are not guaranteed. Of course, there are other methods for calculating differentials, but they tend to be ad-hoc or context specific.

The best overall summary of results would be Doug Ravenel's book on the homotopy groups of spheres, and I would also recommend Kochman's book. Read works of Mark Mahowald for results using the Adams spectral sequence, and Doug Ravenel for the Adams-Novikov spectral sequence. Complete or nearly complete calculations for the homotopy groups of spheres that have been localized at a particular Morava K-theory have been made by Toda, Goerss-Henn-Mahowald-Rezk, and Mark Behrens. If you're interested in computer calculations of Ext, you should contact Robert Bruner or Christian Nassau. Many others have contributed to the calculation of homotopy groups of spheres and probably deserve to be mentioned (if I omitted someone, it was unintentional).

On an unrelated and personal note: I would like to publicly thank Torsten Ekedahl (who recently passed away) for everything he has done to help me.