In my research, it so happens that it might be useful to look at foliations on a surface. I am however having a hard time absorbing a new subject. So, could somebody point me to some suitable references for the simpler situation of foliations on the real line $\mathbb R$, and their structural properties, classification etc?
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I am not sure that your question makes sense.
A foliation on a surface decomposes the surface continuously into one-dimensional "leaves". (Think of taking a vector field on your surface; the corresponding flow will give you a foliation.) The simplest example of a foliation is probably to form a torus by identifying opposite sides of a unit square. Take an angle $\alpha$ and look at all the lines that have angle $\alpha$ with the real axis; these will project down to a foliation. If the slope of the lines is rational, then the corresponding foliation will consist of closed curves; otherwise, however, each leaf keeps winding around the torus.
More generally, you will want to allow some isolated singularities in your foliations, but essentially the picture is as described above.
If you were going to consider foliations of the real line, then the leaves would be zero-dimensional, i.e. they would just be points - so there is just one "foliation", namely the decomposition of the real line into singletons!
So I'm afraid you'll have to stick with foliations on surfaces, but that's not so bad; after all, in two dimensions you can easily draw pictures to get some idea of what is going on.
Instead of looking at foliations of 1-dimensional spaces, look at foliations of 2-dimensional spaces with 1-dimensional leaves. That is, look at foliations of the plane by curves. A simple family of examples is given by the level curves of a function $f:\mathbb R^2 \to \mathbb R$. Not all foliations are like that but it helps to start getting some intuition. More generally, look at the solutions of 2-dimensional autonomous ordinary differential equations. The level curve example comes from Hamiltonians equations.
For a book, try Geometric Theory of Foliations by Camacho and Lins Neto.
I agree with mathstribble that a foliation of $\mathbb R$ are just the set of points of $\mathbb R$ (which of course is $\mathbb R$ itself) and not of interest. However, since what you really wish to understand is foliations of surfaces, I'll point you to a paper that may help: H. B. Lawson. Foliations. Bull. Amer. Math. Soc. 80:3 (1974), 369–418. MR 0343289 (49 #8031).