Vasileios,
By doing only one contour you can convert it to points. You should use Normal to convert the ContourPlot format from GraphicsComplex to Graphics. Then you can extract the lines by searching for them.
For your example, I edited your example by deleting all but one contour value ({0.8}:
oneContour =
Show[ContourPlot[
Simplify[
u^2 + d/(u*R^2)^(c - 1) - a/R +
b (2 (1 - R^2)/(1 - R^2 u) + (R^2 -
R^2 u)^2/(R^2 (1 - R^2 u)^2)) - (Bernoullifast +
Bernoullislow)/2 /. c -> c1[[1]] /. a -> a1[[1]] /.
b -> b1[[1]]], {R, 0.0, 1.0}, {u, 0.0, 3.0},
ContourShading -> None, PlotPoints -> 100, Contours -> {0.8}]
, FrameLabel -> {{HoldForm[u], None}, {HoldForm[R], None}},
LabelStyle -> {GrayLevel[0]}] /.
Tooltip[x_, 0.028] :> Tooltip[newStyle[x], 0.028]
To get the points (and then replot them)
ListPlot[Cases[Normal[oneContour], Line[x_] :> x, Infinity]]
Regards,
Neil