tl;dr - The equations you are looking for are the Navier-Stokes equations. The Reynolds number is a convenient non-dimensionalizing parameter in these equations. The Navier-Stokes equations are hard to solve, so there isn't an easy way to find drag (or lift) as a function of Reynolds number.
From the Wikipedia article for Reynolds number:
In fluid mechanics, the Reynolds number ($Re$) is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions.
In addition, the incompressible Navier-Stokes equations, which govern continuum fluid flow, can be written in non-dimensional form such that the only parameter is the Reynolds number (ignoring body forces). This is very nice because it is the basis for the validity of wind tunnel testing.
Suppose we would like to measure the aerodynamics of the flow around a Boeing 747 that is landing. Two (at least) options exist:
- Build your very own full size 747, instrument it, and fly it. (extremely expensive)
- Build a small scale model of a 747, instrument it, test inside a wind tunnel (much less expensive)
But how do we know that the flow we measure in the wind tunnel is what really happens in flight? We match the Reynolds numbers and the exact same equations model both situations--therefore the aerodynamics must be the same.
You have not seen any equations that use the Reynolds number to calculate lift, drag, etc, because the relationship between the Reynolds number and these values is very complicated. The only equations that can fully capture this are the Navier-Stokes equations.
Lift is not affected too much by Reynolds number but drag is. Generally speaking, the higher the Reynolds number, the more likely the flow over a surface is turbulent. The drag associated with turbulent boundary layers is much higher than laminar boundary layers. As a result, drag is incredibly sensitive to the location where the boundary layer transitions from laminar to turbulent flow.
Despite nearly a century of work on laminar-turbulent transition, a reliable method to predict the location of transition in all cases has not been found.