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The following works to give the title all in black:

title{The Title}

But this does not work to make the first word in the title red:

title{{color{red}The} Title}

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).