How does one make just one word in the title a different color?

The \title under amsart is set in CAPITAL LETTERS. As such, merely using

\title{\textcolor{red}{The} title}

doesn't work out-of-the-box because there is no colour RED. However, it's simple enough to define RED to be the same as red:

\documentclass{amsart}

\usepackage{xcolor}

\colorlet{RED}{red}
\title{\textcolor{red}{The} title}
\author{An Author}

\begin{document}

\maketitle

\end{document}

With amsart there is the problem of capitalization, but it is easily solved.

\documentclass{amsart}
\usepackage{xcolor}
\usepackage{textcase}

\DeclareRobustCommand{\foo}[1]{\textcolor{red}{#1}}

\begin{document}

\title[The title]{\foo{The} title}
\author{An Author}

\maketitle

\end{document}

Use a more meaningful name than \foo, of course. Omit the optional argument if you also want the coloring in the page headings.

See https://tex.stackexchange.com/a/468247/4427 for details about why loading textcase is better for the application.

The rationale for defining a command is that hardwiring a color means having to chase for every occurrence in the document for it, whereas with a command you can just modify the definition in case you change your mind about what color to use (or no color at all).