The problem is that the spanish module for babel makes > and < active characters for its special management of Spanish quotations.
Update:
With PGF/TikZ version 3.0.0 there's a babel library that solves the problems produced by changes to category codes made by some babel modules, so it's enough to load the library:
\documentclass[spanish]{article}
\usepackage{babel}
\usepackage{tikz}
\usetikzlibrary{babel}
\begin{document}
Simple arrows:
\begin{tikzpicture}
\draw[->] (-1.5,0) -- (1.5,0);
\end{tikzpicture}
\tikz\draw[->] (-1.5,0) -- (1.5,0);
\end{document}
(Thanks to Claudio Fiandrino for pointing this out in his comment).
PGF/TikZ version 2.10
To deactivate the undesired redefinition for tikzpictures, but maintaining it for other parts of the document, you can use
execute at begin picture={\deactivatequoting},
execute at end picture={\activatequoting}
for every tikzpicture. Notice that this approach has two advantages:
No extra packages are required.
It also produces the desired result when using the command version \tikz (see example code below).
The code:
\documentclass[spanish]{article}
\usepackage{babel}
\usepackage{tikz}
\tikzset{
every picture/.append style={
execute at begin picture={\deactivatequoting},
execute at end picture={\activatequoting}
}
}
\begin{document}
Simple arrows:
\begin{tikzpicture}
\draw[->] (-1.5,0) -- (1.5,0);
\end{tikzpicture}
\tikz\draw[->] (-1.5,0) -- (1.5,0);
\end{document}

As a side note, don't use pdftex option; modern LaTeX systems detect the driver automatically.
\usepackage[english]{babel}works, but\usepackage[spanish]{babel}does not. Also, not sure why you have[pdftex,spanish]options for\documentclass- FYI, I am not that familiar withbabel.\shorthandoff{>}\shorthandoff{<}after\begin{document}. < and > are active characters in Spanish.pdftexoption is not necessary with modern LaTeX systems (the driver is automatically detected). Using the idiomatic localization (spanish, in this case) as class option has the advantage that all language-sensitive packages (for example,babel,cleveref) that can eventually be loaded will pick it and produce the idiomatic localizations.