The formatting of the names used in the citation call-outs is governed not by the natbib package but by the bibliography style that's in use.
I'm assuming that you'll want to show the first names of all authors, rather than just those of of selected authors. While you're at it, I'll also assume that you'd want to show any "junior" name components, if present. Finally, you'll want the entries still sorted by author's surnames rather than by their first names, right?
Since you're using the plainnat bibliography style, I suggest you proceed as follows:
Find the file plainnat.bst in your TeX distribution and make a copy of this file. Call the copy, say, plainnat-full.bst. (Don't edit or modify a file from the TeX distribution directly.)
Open the file plainnat-full.bst in a text editor. The editor you use for your tex files will do fine.
In plainnat-full.bst, locate the function format.lab.names. (In my copy of the the file, this function starts on line 1101.)
In this function, replace the line
s #1 "{vv~}{ll}" format.name$
with
s #1 "{ff }{vv~}{ll}{, jj}" format.name$
Still in the same function, replace the line
{ " and " * s #2 "{vv~}{ll}" format.name$ * }
with
{ " and " * s #2 "{ff }{vv~}{ll}{, jj}" format.name$ * }
If you don't want to show the "junior" components, leave off the {, jj} substrings in the preceding two steps.
Save the file plainnat-full.bst, and store it either in the directory that contains your main .tex file or in a directory that's searched by BibTeX. If you select the latter option, be sure to update the filename database of your TeX distribution as well.
In your main tex file, change the instruction \bibliographystyle{plainnat} to \bibliographystyle{plainnat-full} and recompile the document by re-running LaTeX, BibTeX, and LaTeX twice more to fully propagate all changes.
Happy BibTeXing!

\RequirePackage{filecontents}
\begin{filecontents}{bibx.bib}
@misc{smith:3001,
author = "Smith, Jr., John and Smythe, III, Anna",
title = "Thoughts",
year = 3001,
}
\end{filecontents}
\documentclass{article}
\usepackage{natbib}
\bibliographystyle{plainnat-full}
\begin{document}
\citet{smith:3001}
\citep{smith:3001}
\bibliography{bibx}
\end{document}