Where does classical physics end and where does modern physics begin?

I would say that Einstein was the dividing line.  Of course it didn't change overnight but it all changed after Einstein.  Changed again with quantum physics but our understanding of things is always changing.

Read Korzybski's Science and Sanity, and some of the writings from the Null-A society. The real difference between modern science and classical science is that classical science took Euclid's five postulates as being the minimum neceessary. Modern stuff only uses four postulates for geometry, making Euclidean geometry a special case.

The classicists thought that mass was always conserved, and that energy was always (well mostly) conserved. The two were very separate considerations. With Einstein, and with quantum mechanics, energy-mass (the two together) are conserved, so that mass can become energy, and energy can become (or have) mass.

So, while classical mechanics is still regarded as holding "locally", it doesn't handle things like the speed of light being a constant (in vacuum), or that mass of a body travelling at speeds close to light speed have a different mass (if you stretch things, the Euclidean/Newtonian approach can seem to fit the physics, but it is a real stretch).

when you think of natural world,it`s classical physics,when you begin to babble about black and white and worm holes,parallel lines of infinity,string laces and chopping materials to the core,it`s `modern`...

The accepted position is that the quantum Physics is the ultimate theory of matter which reduces to classical Physics under certain circumstances which are well defined. There is a Bohr's Correspondence principle to take care of that. Similarly Theory of relativity is the ultimate theory of space time and it reduces to classical theory when velocities are much below that of velocity of light in vacuum. Complete unification of these two is yet not complete I think. The stuff which you call classical physics is the use of mathematical concepts. These concepts are useful in quantum physics as well. Electromagnetic theory is a classical theory and has no place for the concept of photon or inter-conversion of matter into radiation and vice-a-versa. Blend of the two is sometimes easy to handle but is not equally powerful as the pure quantum theory, still this blend or semi-quantum theory explains things which pure classical theory cannot dream of explaining.

Classical and quantum considerations in the fundamental sense are poles apart. But both have ideas not akin to common sense. Classical physics start with the concept of point particle with so called forces (fields) between them and builds a theoretical structure using mathematical idea. point particle and force are both abstractions. Quantum field theory talks about fields dependent on exchange of particles which need not be point particles.

when the number of particles is high and there is some kind of randomization, probability and statistical concepts need to be used. So we have both Classical statistical physics as well as Quantum statistics. However at the fundamental level classical laws are deterministic, where as quantum laws are probabilistic. However contrary to common belief there is more experimental evidence from solid state and modern physics in favour of quantum physics and relativity, such evidence which cannot be explained by classical ideas.

In a lighter vein, quantum physics does not pretend to know what cannot be known! If we do not see the electron revolving around the nucleus in an atom, then it does not assume this or propose a hypothesis for it. However it cannot make a beginning in vacuum. For that it heavily depends on the theoretical structure available with classical theory rather it begins with that.