I'm a bit stuck on the general concept and calculations of  iterated expectations. A simple example:

$$E[E[Y|X]] = E[Y]$$

I'm not sure how or why this is the case? I have a 3 line proof in my notes using densities and stuff but is there no 'quicker' way. Especially in longer cases like (below) writing everything out is difficult. Are there tricks to spot? 

Example 2:

$Var[Y] = E[Y^2] - (E[Y])^2$ (I get this line!)
$$= E[E[Y^2|X]] - \bigg(E[E[Y|X]]\bigg)^2 $$
$$= E\left[Var[Y|X]+(E[Y|X])^2\right] - \bigg(E[E[Y|X]]\bigg)^2$$
$$= E[Var[Y|X]] + E[(E[Y|X])^2] - \bigg(E[E[Y|X]]\bigg)^2$$
$$= E[Var[Y|X]] + Var[E[Y|X]]$$

I'm not at all sure how they go from line to line.