The first thing to do would be to check if your distribution doesn't already provide a way to add the package to your TeX installation. You didn't say which distribution you used (MacTeX ? TeX Live ?) and that's actually the most important part. The editor you use for typing source files is unrelated, it's only the front-end to the actual TeX binaries and files.
To answer your second question, you do need core PSTricks macros in order to use any PSTricks-related package, so do that first; all the major distributions ship PSTricks, so first try to figure out how to install it (and maybe it's there already). There really should be a way to do so, so only try the steps in the next paragraphs if all else fails.
If installation is not possible via graphical applications shipped with your TeX distribution, you may have to install the files by hand. I outline the process briefly here, but you can read the manual for "kpathsea" for more information (it's the library through which TeX and related programs find files). First identify the main directory where your TeX files resides; it's usually called texmf-[something]. In TeX Live it's /usr/local/texlive/[year]/texmf-dist by default; in MacTeX it should be somewhere under /Library/TeX/Distributions.
From there, look for a directory called texmf-local (or texmf.local, etc.); that's where you should drop the files; more precisely in texmf-local/tex/latex/pst-gantt. Then run "texhash" from the command line in Terminal, and you should be done. The latter command may not be in your PATH environment variable, so you may have to set up this first.
Hope this helps.
Oh, and you don't need .dtx and .ins; they're just a LaTeXy way of packing all the files together for upstream distribution; what you really need is the .sty file, as well as, obviously, docs.