Is $3^x \lt 1 + 2^x + 3^x \lt 3 \cdot 3^x$ right? Is $3^x \lt 1 + 2^x + 3^x \lt 3 \cdot 3^x$ right?
This is from my lecture notes which is used to solve: 


But when $x = 0$, $(1 + 2^x + 3^x = 3) \gt (3^0 = 1)$? The thing is how do I choose which what expression should go on the left & right side? 

Firstly, your lecture notes says "For all $x>0$", so you can't choose $0$. In fact:
Let $I$ be an interval having the point $a$ as a limit point. Let $f$, $g$, and $h$ be functions defined on $I$, except possibly at a itself. Suppose that for every $x$ in $I$ not equal to a, we have:
$$g(x)\leq f(x)\leq h(x)$$
also suppose that
$$\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L$$
Then we have
$$\lim_{x\to a}f(x)=L$$
So it doesn't necessary have to be "For all $x>0$". If you know Your limit points is $+\infty$ so all you need is to be sure that some relation
$$g(x)\leq f(x)\leq h(x)$$
holds starting with some $x$.
