TITLE: Elementary Morse theory
QUESTION [9 upvotes]: I want to initiate myself to 'elementary' Morse theory and use it to calculate the Euler-Poincare characteristic of some compact manifolds (spheres and torus ...).
I do not know what strategy should I follow:

Given a compact submanifold $M$ of $\mathbb{R}^n$, if I take the function distance from a point $p$, $\|x-p\|$ (I have to check from the Sard's Theorem that its a Morse function for almost all $p$). Otherwise, if it is not a Morse function, how to 'deform it' to obtain a Morse one?
If I take the height function $f(x,y,z)=z$ on the sphere $S^2$, it admits two critical points $(0,0,-1)$ and $(0,0,1)$ (corresponding to the min and max of $f$). How to use the Morse lemma to determine the index of each critical point, then the Euler-Poincare characteristic?
How to show that the Euler characteristic does not depend on the choice of a particular Morse function?
Why the concept of transversality is important in Morse theory?
too many questions ...! I thank everyone telling me some ideas and some good introductory material.

REPLY [3 votes]: A couple of years ago, Alexander Ritter taught a course in Morse homology whose lectures notes can be found here. They take a modern perspective, and so are quite heavy on functional analysis.
In any case, there should be something of use there, in particular, they answer your questions about independence of choice of Morse function and where transversality comes into the picture.