How do I put big brackets under different parts of an equation to attach text labels?

What you want is

f = underbrace{x^3}_textrm{text 1} + underbrace{2}_textrm{text 2}

You would find it out in the mathmode manual or symbols manual.

Since the text used in underbrace (or overbrace) is typically set in a different font size to not distract further from the equation, it is preferable to use amsmath's text macro.

documentclass{article}
% usepackage{amsmath}
usepackage{mathtools}% Loads amsmath
begin{document}
[
  f(x) = 
    underbrace{(x + 2)^3}_text{text 1} + 
    bigl(
      mathrlap{overbrace{phantom{(c - 2d)}}^{text{text 2}}}
      (c - 
      mathrlap{underbrace{phantom{2d) + (3e}}_{text{text 3}}}
      2d) +
      overbrace{(3e - 4f)}^{text{text 4}}
    bigr) + 
    overbrace{(x - 3)}^text{text 5}
]
end{document}

The difficult overlapping braces uses math overlaps from mathtools. It follows a process of setting the over-/underbrace text first with a phantom base, after which the base (or part thereof) is re-set. However, if those aren't needed, using amsmath only would suffice for usage of text.

There are a number of ways of achieving the overlapping output. Another method of overlapping braces is discussed in section 63.2 Overlapping braces of the mathmode document.