Should i care about the previous versions?
Yes. You should at least know which features of 4.0 the older version DON'T have.
There's a good chance that if you're using .NET on the job, they're not going to be at 4.0 yet. Therefore, you're going to need to know which features you can use and which you're going to have to work around.
What are the tips you give for beginner to learn c# quickly?
Find a good book and start at the beginning.
I would recommend either Pro C# 2010 and the .NET 4.0 Platform or C# In Depth: Second Edition
What are the factors to be considered when moving from c,c++ to c#?
Know the naming conventions and differences of each.
C/C++ gets compiled to native code. C# gets JIT compilation.
C# has memory management. C/C++ doesn't (at least not the same).
C is not an object oriented langauge. C++/C# are, but each handles OO in a slightly different way.
Those kind of differences will change how you write/think about your code.