Changed answer
The ligatures are used by default, however that do not look very much like ligatures

\documentclass{article}
\usepackage{mathpazo}
\usepackage[T1]{fontenc}
\begin{document}
fi ffi
\end{document}
(The palatino package is obselete, as the documentation psnfss2e.pdf will tell you.)
Inserting
\usepackage[loading]{tracefnt}
the log file will tell you near the end
LaTeX Font Info: External font `pplr8t at10.0pt' loaded as`
(Font) T1/ppl/m/n/10 on input line 7.`
Running
\documentclass{article}
\usepackage{fonttable}
\usepackage[T1]{fontenc}
\begin{document}
\fonttable{pplr8t}
\end{document}
through pdflatex will give a table of characters in the font and you will see there are ligatures provided for ff, fi, fl, ffi and ffl - but they look very much like the composite letters (the most obvious difference is the reduced space in the ff combinations compared to f\/f).
Furthermore it should be mentioned that there are other fonts in the palatino bundle under psnsfss2e. mathpazo can be loaded with the [sc] option or the [osf] option. The first provides true small caps, the second provides additionally old style numbers.
Finally you can omit the T1 fontenc and get OT1 instead, but psnfss2e.pdf recommands T1.
In all cases the ligatures in the corresponding font tables appear the same - the psnfss2e documentation says that all are Palatino-Roman.