What is needed to go from a student pilot license to a private pilot license?
You have a bit of a misconception about the way these things work.  Everyone gets a student pilot's license before they get a private pilot's license.  The student license is just a step along the way to the private license.  The requirements for getting a student pilot's license are minimal.  It's the equivalent of a "learner's permit" in the automobile world.  The reason that the requirements for a student pilot's license are minimal is that you will be operating under the supervision of an instructor.  Just having the student license does not authorize you to fly an airplane without an instructor on board.  When you hold a student pilot's license, an instructor has to endorse your license for solo privileges in each make and model of aircraft, and also has to endorse your logbook for solo privileges every ninety days.  An instructor also has to review your planning for every cross-country flight, with exceptions for flights to another airport within 25 miles or repeated flights to another airport within 50 miles.  These and other requirements for solo flight under a student pilot's license are spelled out in FAR 61.87 and 61.93.

Without a student pilot's license, there would be no way to build solo flight time as needed to qualify to take the checkride for a private pilot's license.

This answer is assuming that

1) You live in the USA

2) You haven't already learned to fly, and accumulated flight time, in the military..

Highly related-- essentially the same question in fact-- see all answers--

Am I required to obtain a student license before a sport, recreational, or private one?