I want to know what all doubly-transitive groups look like. Do you know some good reference where I can read about it?
In Section 7.7 "The Finite 2-transitive Groups" of the book Permutation groups by John D. Dixon and Brian Mortimer, the authors describe the complete list of finite 2-transitive groups without proofs but with references.
They list eight infinite families: the alternating, symmetric, affine and projective groups in their natural actions, as well as the less known groups of Lie type: the symplectic groups, the Suzuki groups, the unitary groups and the Ree groups. The symplectic groups have two distinct 2-transitive actions, the last three classes are 2-transitive on the sets of points in their action on appropriate Steiner systems. Additional there are 10 sporadic examples of 2-transitive groups.
The solvable case was proved by Huppert in 1957. It is presented in textbook form in chapter XII.7 of Huppert–Blackburn's Finite Groups Vol. 3. This includes full proof that only certain exceptional cases can occur, but not full coverage of the exceptional cases. It includes an outline of Hering (et al.)'s proof of the insoluble case, which is spread over several papers (which are the best source of the proof).
The solvable case is not classical and is not presented in Robinson's textbook. Presumably people are thinking of solvable groups of prime degree. Cameron's book does not contain any proof, and its tables are partial so be careful of relying on it for details. Obviously the classical textbooks can only present partial pictures as the results were not known at the time.