I am by no means a LaTex expert (and there are some here, such as Will, who are) but I do routinely use MikTex on Windows and I love the results, so I’ll try to tell you what works for me.
For an editor, I use Texmaker (http://www.xm1math.net/texmaker/) which has always worked great for me. I’m sure there are all kinds of discussions floating around the internet about which is the best, but this one seems to handle unicode very well and has never given me any problems.
The key, at least for me, to getting it to work was to add the command for xelatex (the version of latex that handles unicode well). To set this up, I go to the menu option “User” and then choose " For “Command 1” I give it the name “XeTex” and put in this command:
xelatex -interaction=nonstopmode %.tex
Once this has been set, exit then restart texmaker. Then, along the top row where the buttons are, there is one that defaults to something like “Quick build.” I click on the little down arrowhead and change it to 1: XeTex from the drop-down menu. At this point, any xetex file will compile fine for me and generate a pdf file simply by pressing the arrow button just to the left of the “1: XeTex” text. Usually the first time I do this on a computer, there will be a bunch of packages that get installed. Here’s a simple xetex file that illustrates how I change fonts:
\documentclass{article}
\usepackage{fontspec}
\usepackage{xunicode}
\usepackage{xltxtra}
\setmainfont[Mapping=tex-text]{KadmosU}
\begin{document}
\section{Unicode}
\subsection{Kadmos}
ὁ Βορέας ποτὲ καὶ ὁ ἥλιος ἤριζον ὁπότερός ἐστι κρείσσων. ἰδὼν δὲ ἄνθρωπον ἱμάτιον ἔχοντα, ἔ-θη ὁ ἥλιος.
“τί μάτηρ διαλεγόμεθα; ὁρᾷς δήπου ἐκεῖνον τὸν ἄνδρα· ὁποτερος ἂν οὖν ἡμῶν δύνηται ἀποδῦσαι τὸ ἱμάτιον τοῦ ἀνθρώπου,
οὖτος νικάτω.” ὁ δὲ ἕτερος ἐπὶ τούτοις ὡμολόγησε. πρῶτον μὲν οὖν πεῖραν ἐποιεῖτο ὁ Βορέας. καὶ παντὶ μὲν τῷ σθένει ἔ-πνευσε,
ὁ δὲ ἀνὴρ πολὺ μᾶλλον ἑαυτὸν τῷ ἱματίῳ ἐ-κάλυπτε. παυσαμένου δὲ τοῦ Βορέου, ὁ ἥλιος ἐξ-ῆλθεν. ὁ δὲ ἀνήρ, θερμὸς γενόμενος, τὸ
ἱμάτιον ἀπ-έ-ρριψε.
\font\1="Gentium" at 12pt
\subsection{Gentium}
\1
ὁ Βορέας ποτὲ καὶ ὁ ἥλιος ἤριζον ὁπότερός ἐστι κρείσσων. ἰδὼν δὲ ἄνθρωπον ἱμάτιον ἔχοντα, ἔ-θη ὁ ἥλιος.
“τί μάτηρ διαλεγόμεθα; ὁρᾷς δήπου ἐκεῖνον τὸν ἄνδρα· ὁποτερος ἂν οὖν ἡμῶν δύνηται ἀποδῦσαι τὸ ἱμάτιον τοῦ ἀνθρώπου,
οὖτος νικάτω.” ὁ δὲ ἕτερος ἐπὶ τούτοις ὡμολόγησε. πρῶτον μὲν οὖν πεῖραν ἐποιεῖτο ὁ Βορέας. καὶ παντὶ μὲν τῷ σθένει ἔ-πνευσε,
ὁ δὲ ἀνὴρ πολὺ μᾶλλον ἑαυτὸν τῷ ἱματίῳ ἐ-κάλυπτε. παυσαμένου δὲ τοῦ Βορέου, ὁ ἥλιος ἐξ-ῆλθεν. ὁ δὲ ἀνήρ, θερμὸς γενόμενος, τὸ
ἱμάτιον ἀπ-έ-ρριψε.
\end{document}
(for this to work, your computer would have to have both Gentium and KadmosU fonts installed (both are free out on the internet) but obviously you could replace them with whatever fonts you choose. What I did in this file is make KadmosU the default font for the document (up near the beginning of the document) so the first paragraph is done in this font. Then I switch to the Gentium font with the \1 business.
Anyway, I certainly don’t know the finer points of all of this stuff, but this technique has served me well and generates beautiful pdf files. I’m sure there are other slicker ways to do all this stuff, and maybe someone else will chime in, but at least this is a start.
As an aside, a ton of things online tell you to install all kinds of other stuff, but all I ever need to install are MikTex and Texmaker. Those two alone take care of everything, and Texmaker has always been smart enough to figure out where MikTek is. Good luck and I hope all this works for you. If you have trouble, I’ll try to help.